registered buffers' dtype is overridden after __init__

Nobody has claimed this yet.

### Bug description

If I register a `float64` tensor to a buffer in the `__init__` function of a `LightningModule` like so:

```py
self.register_buffer("testing_variable", torch.tensor([1,2,3], dtype=torch.float64))
```

It will get cast into the `Trainer`'s `precision` type after setup, regardless of whether a different dtype (e.g. `float64`) was intended.

### What version are you seeing the problem on?

v2.1

### How to reproduce the bug

https://colab.research.google.com/drive/1UN1JeuiV3LM-LRrATigboJeDH9ehJifY

### Error messages and logs

N/A

### Environment

Current environment

* CUDA:
- GPU:
- Tesla T4
- available: True
- version: 11.8
* Lightning:
- lightning: 2.1.1
- lightning-utilities: 0.9.0
- pytorch-lightning: 2.1.1
- torch: 2.1.0+cu118
- torchaudio: 2.1.0+cu118
- torchdata: 0.7.0
- torchmetrics: 1.2.0
- torchsummary: 1.5.1
- torchtext: 0.16.0
- torchvision: 0.16.0+cu118
* Packages:
- absl-py: 1.4.0
- aiohttp: 3.8.6
- aiosignal: 1.3.1
- alabaster: 0.7.13
- albumentations: 1.3.1
- altair: 4.2.2
- anyio: 3.7.1
- appdirs: 1.4.4
- argon2-cffi: 23.1.0
- argon2-cffi-bindings: 21.2.0
- array-record: 0.5.0
- arviz: 0.15.1
- astropy: 5.3.4
- astunparse: 1.6.3
- async-timeout: 4.0.3
- atpublic: 4.0
- attrs: 23.1.0
- audioread: 3.0.1
- autograd: 1.6.2
- babel: 2.13.1
- backcall: 0.2.0
- beautifulsoup4: 4.11.2
- bidict: 0.22.1
- bigframes: 0.12.0
- bleach: 6.1.0
- blinker: 1.4
- blis: 0.7.11
- blosc2: 2.0.0
- bokeh: 3.3.0
- bqplot: 0.12.42
- branca: 0.7.0
- build: 1.0.3
- cachecontrol: 0.13.1
- cachetools: 5.3.2
- catalogue: 2.0.10
- certifi: 2023.7.22
- cffi: 1.16.0
- chardet: 5.2.0
- charset-normalizer: 3.3.2
- chex: 0.1.7
- click: 8.1.7
- click-plugins: 1.1.1
- cligj: 0.7.2
- cloudpickle: 2.2.1
- cmake: 3.27.7
- cmdstanpy: 1.2.0
- colorcet: 3.0.1
- colorlover: 0.3.0
- colour: 0.1.5
- community: 1.0.0b1
- confection: 0.1.3
- cons: 0.4.6
- contextlib2: 21.6.0
- contourpy: 1.2.0
- cryptography: 41.0.5
- cufflinks: 0.17.3
- cupy-cuda11x: 11.0.0
- cvxopt: 1.3.2
- cvxpy: 1.3.2
- cycler: 0.12.1
- cymem: 2.0.8
- cython: 3.0.5
- dask: 2023.8.1
- datascience: 0.17.6
- db-dtypes: 1.1.1
- dbus-python: 1.2.18
- debugpy: 1.6.6
- decorator: 4.4.2
- defusedxml: 0.7.1
- diskcache: 5.6.3
- distributed: 2023.8.1
- distro: 1.7.0
- dlib: 19.24.2
- dm-tree: 0.1.8
- docutils: 0.18.1
- dopamine-rl: 4.0.6
- duckdb: 0.9.1
- earthengine-api: 0.1.377
- easydict: 1.11
- ecos: 2.0.12
- editdistance: 0.6.2
- eerepr: 0.0.4
- en-core-web-sm: 3.6.0
- entrypoints: 0.4
- et-xmlfile: 1.1.0
- etils: 1.5.2
- etuples: 0.3.9
- exceptiongroup: 1.1.3
- fastai: 2.7.13
- fastcore: 1.5.29
- fastdownload: 0.0.7
- fastjsonschema: 2.18.1
- fastprogress: 1.0.3
- fastrlock: 0.8.2
- filelock: 3.13.1
- fiona: 1.9.5
- firebase-admin: 5.3.0
- flask: 2.2.5
- flatbuffers: 23.5.26
- flax: 0.7.5
- folium: 0.14.0
- fonttools: 4.44.0
- frozendict: 2.3.8
- frozenlist: 1.4.0
- fsspec: 2023.6.0
- future: 0.18.3
- gast: 0.5.4
- gcsfs: 2023.6.0
- gdal: 3.4.3
- gdown: 4.6.6
- geemap: 0.28.2
- gensim: 4.3.2
- geocoder: 1.38.1
- geographiclib: 2.0
- geopandas: 0.13.2
- geopy: 2.3.0
- gin-config: 0.5.0
- glob2: 0.7
- google: 2.0.3
- google-api-core: 2.11.1
- google-api-python-client: 2.84.0
- google-auth: 2.17.3
- google-auth-httplib2: 0.1.1
- google-auth-oauthlib: 1.0.0
- google-cloud-bigquery: 3.12.0
- google-cloud-bigquery-connection: 1.12.1
- google-cloud-bigquery-storage: 2.22.0
- google-cloud-core: 2.3.3
- google-cloud-datastore: 2.15.2
- google-cloud-firestore: 2.11.1
- google-cloud-functions: 1.13.3
- google-cloud-iam: 2.12.2
- google-cloud-language: 2.9.1
- google-cloud-resource-manager: 1.10.4
- google-cloud-storage: 2.8.0
- google-cloud-translate: 3.11.3
- google-colab: 1.0.0
- google-crc32c: 1.5.0
- google-pasta: 0.2.0
- google-resumable-media: 2.6.0
- googleapis-common-protos: 1.61.0
- googledrivedownloader: 0.4
- graphviz: 0.20.1
- greenlet: 3.0.1
- grpc-google-iam-v1: 0.12.6
- grpcio: 1.59.2
- grpcio-status: 1.48.2
- gspread: 3.4.2
- gspread-dataframe: 3.3.1
- gym: 0.25.2
- gym-notices: 0.0.8
- h5netcdf: 1.3.0
- h5py: 3.9.0
- holidays: 0.36
- holoviews: 1.17.1
- html5lib: 1.1
- httpimport: 1.3.1
- httplib2: 0.22.0
- humanize: 4.7.0
- hyperopt: 0.2.7
- ibis-framework: 6.2.0
- idna: 3.4
- imageio: 2.31.6
- imageio-ffmpeg: 0.4.9
- imagesize: 1.4.1
- imbalanced-learn: 0.10.1
- imgaug: 0.4.0
- importlib-metadata: 6.8.0
- importlib-resources: 6.1.1
- imutils: 0.5.4
- inflect: 7.0.0
- iniconfig: 2.0.0
- install: 1.3.5
- intel-openmp: 2023.2.0
- ipyevents: 2.0.2
- ipyfilechooser: 0.6.0
- ipykernel: 5.5.6
- ipyleaflet: 0.17.4
- ipython: 7.34.0
- ipython-genutils: 0.2.0
- ipython-sql: 0.5.0
- ipytree: 0.2.2
- ipywidgets: 7.7.1
- itsdangerous: 2.1.2
- jax: 0.4.20
- jaxlib: 0.4.20+cuda11.cudnn86
- jeepney: 0.7.1
- jieba: 0.42.1
- jinja2: 3.1.2
- joblib: 1.3.2
- jsonpickle: 3.0.2
- jsonschema: 4.19.2
- jsonschema-specifications: 2023.7.1
- jupyter-client: 6.1.12
- jupyter-console: 6.1.0
- jupyter-core: 5.5.0
- jupyter-server: 1.24.0
- jupyterlab-pygments: 0.2.2
- jupyterlab-widgets: 3.0.9
- kaggle: 1.5.16
- keras: 2.14.0
- keyring: 23.5.0
- kiwisolver: 1.4.5
- langcodes: 3.3.0
- launchpadlib: 1.10.16
- lazr.restfulclient: 0.14.4
- lazr.uri: 1.0.6
- lazy-loader: 0.3
- libclang: 16.0.6
- librosa: 0.10.1
- lida: 0.0.10
- lightgbm: 4.1.0
- lightning: 2.1.1
- lightning-utilities: 0.9.0
- linkify-it-py: 2.0.2
- llmx: 0.0.15a0
- llvmlite: 0.41.1
- locket: 1.0.0
- logical-unification: 0.4.6
- lxml: 4.9.3
- malloy: 2023.1064
- markdown: 3.5.1
- markdown-it-py: 3.0.0
- markupsafe: 2.1.3
- matplotlib: 3.7.1
- matplotlib-inline: 0.1.6
- matplotlib-venn: 0.11.9
- mdit-py-plugins: 0.4.0
- mdurl: 0.1.2
- minikanren: 1.0.3
- missingno: 0.5.2
- mistune: 0.8.4
- mizani: 0.9.3
- mkl: 2023.2.0
- ml-dtypes: 0.2.0
- mlxtend: 0.22.0
- more-itertools: 10.1.0
- moviepy: 1.0.3
- mpmath: 1.3.0
- msgpack: 1.0.7
- multidict: 6.0.4
- multipledispatch: 1.0.0
- multitasking: 0.0.11
- murmurhash: 1.0.10
- music21: 9.1.0
- natsort: 8.4.0
- nbclassic: 1.0.0
- nbclient: 0.9.0
- nbconvert: 6.5.4
- nbformat: 5.9.2
- nest-asyncio: 1.5.8
- networkx: 3.2.1
- nibabel: 4.0.2
- nltk: 3.8.1
- notebook: 6.5.5
- notebook-shim: 0.2.3
- numba: 0.58.1
- numexpr: 2.8.7
- numpy: 1.23.5
- oauth2client: 4.1.3
- oauthlib: 3.2.2
- opencv-contrib-python: 4.8.0.76
- opencv-python: 4.8.0.76
- opencv-python-headless: 4.8.1.78
- openpyxl: 3.1.2
- opt-einsum: 3.3.0
- optax: 0.1.7
- orbax-checkpoint: 0.4.2
- osqp: 0.6.2.post8
- packaging: 23.2
- pandas: 1.5.3
- pandas-datareader: 0.10.0
- pandas-gbq: 0.17.9
- pandas-stubs: 1.5.3.230304
- pandocfilters: 1.5.0
- panel: 1.3.1
- param: 2.0.0
- parso: 0.8.3
- parsy: 2.1
- partd: 1.4.1
- pathlib: 1.0.1
- pathy: 0.10.3
- patsy: 0.5.3
- peewee: 3.17.0
- pexpect: 4.8.0
- pickleshare: 0.7.5
- pillow: 9.4.0
- pip: 23.1.2
- pip-tools: 6.13.0
- platformdirs: 3.11.0
- plotly: 5.15.0
- plotnine: 0.12.4
- pluggy: 1.3.0
- polars: 0.17.3
- pooch: 1.8.0
- portpicker: 1.5.2
- prefetch-generator: 1.0.3
- preshed: 3.0.9
- prettytable: 3.9.0
- proglog: 0.1.10
- progressbar2: 4.2.0
- prometheus-client: 0.18.0
- promise: 2.3
- prompt-toolkit: 3.0.39
- prophet: 1.1.5
- proto-plus: 1.22.3
- protobuf: 3.20.3
- psutil: 5.9.5
- psycopg2: 2.9.9
- ptyprocess: 0.7.0
- py-cpuinfo: 9.0.0
- py4j: 0.10.9.7
- pyarrow: 9.0.0
- pyasn1: 0.5.0
- pyasn1-modules: 0.3.0
- pycocotools: 2.0.7
- pycparser: 2.21
- pyct: 0.5.0
- pydantic: 1.10.13
- pydata-google-auth: 1.8.2
- pydot: 1.4.2
- pydot-ng: 2.0.0
- pydotplus: 2.0.2
- pydrive: 1.3.1
- pydrive2: 1.6.3
- pyerfa: 2.0.1.1
- pygame: 2.5.2
- pygments: 2.16.1
- pygobject: 3.42.1
- pyjwt: 2.3.0
- pymc: 5.7.2
- pymystem3: 0.2.0
- pyopengl: 3.1.7
- pyopenssl: 23.3.0
- pyparsing: 3.1.1
- pyperclip: 1.8.2
- pyproj: 3.6.1
- pyproject-hooks: 1.0.0
- pyshp: 2.3.1
- pysocks: 1.7.1
- pytensor: 2.14.2
- pytest: 7.4.3
- python-apt: 0.0.0
- python-box: 7.1.1
- python-dateutil: 2.8.2
- python-louvain: 0.16
- python-slugify: 8.0.1
- python-utils: 3.8.1
- pytorch-lightning: 2.1.1
- pytz: 2023.3.post1
- pyviz-comms: 3.0.0
- pywavelets: 1.4.1
- pyyaml: 6.0.1
- pyzmq: 23.2.1
- qdldl: 0.1.7.post0
- qudida: 0.0.4
- ratelim: 0.1.6
- referencing: 0.30.2
- regex: 2023.6.3
- requests: 2.31.0
- requests-oauthlib: 1.3.1
- requirements-parser: 0.5.0
- rich: 13.6.0
- rpds-py: 0.12.0
- rpy2: 3.4.2
- rsa: 4.9
- scikit-image: 0.19.3
- scikit-learn: 1.2.2
- scipy: 1.11.3
- scooby: 0.9.2
- scs: 3.2.3
- seaborn: 0.12.2
- secretstorage: 3.3.1
- send2trash: 1.8.2
- setuptools: 67.7.2
- shapely: 2.0.2
- six: 1.16.0
- sklearn-pandas: 2.2.0
- smart-open: 6.4.0
- sniffio: 1.3.0
- snowballstemmer: 2.2.0
- sortedcontainers: 2.4.0
- soundfile: 0.12.1
- soupsieve: 2.5
- soxr: 0.3.7
- spacy: 3.6.1
- spacy-legacy: 3.0.12
- spacy-loggers: 1.0.5
- sphinx: 5.0.2
- sphinxcontrib-applehelp: 1.0.7
- sphinxcontrib-devhelp: 1.0.5
- sphinxcontrib-htmlhelp: 2.0.4
- sphinxcontrib-jsmath: 1.0.1
- sphinxcontrib-qthelp: 1.0.6
- sphinxcontrib-serializinghtml: 1.1.9
- sqlalchemy: 2.0.23
- sqlglot: 17.16.2
- sqlparse: 0.4.4
- srsly: 2.4.8
- stanio: 0.3.0
- statsmodels: 0.14.0
- sympy: 1.12
- tables: 3.8.0
- tabulate: 0.9.0
- tbb: 2021.10.0
- tblib: 3.0.0
- tenacity: 8.2.3
- tensorboard: 2.14.1
- tensorboard-data-server: 0.7.2
- tensorflow: 2.14.0
- tensorflow-datasets: 4.9.3
- tensorflow-estimator: 2.14.0
- tensorflow-gcs-config: 2.14.0
- tensorflow-hub: 0.15.0
- tensorflow-io-gcs-filesystem: 0.34.0
- tensorflow-metadata: 1.14.0
- tensorflow-probability: 0.22.0
- tensorstore: 0.1.45
- termcolor: 2.3.0
- terminado: 0.17.1
- text-unidecode: 1.3
- textblob: 0.17.1
- tf-slim: 1.1.0
- thinc: 8.1.12
- threadpoolctl: 3.2.0
- tifffile: 2023.9.26
- tinycss2: 1.2.1
- toml: 0.10.2
- tomli: 2.0.1
- toolz: 0.12.0
- torch: 2.1.0+cu118
- torchaudio: 2.1.0+cu118
- torchdata: 0.7.0
- torchmetrics: 1.2.0
- torchsummary: 1.5.1
- torchtext: 0.16.0
- torchvision: 0.16.0+cu118
- tornado: 6.3.2
- tqdm: 4.66.1
- traitlets: 5.7.1
- traittypes: 0.2.1
- triton: 2.1.0
- tweepy: 4.14.0
- typer: 0.9.0
- types-pytz: 2023.3.1.1
- types-setuptools: 68.2.0.0
- typing-extensions: 4.5.0
- tzlocal: 5.2
- uc-micro-py: 1.0.2
- uritemplate: 4.1.1
- urllib3: 2.0.7
- vega-datasets: 0.9.0
- wadllib: 1.3.6
- wasabi: 1.1.2
- wcwidth: 0.2.9
- webcolors: 1.13
- webencodings: 0.5.1
- websocket-client: 1.6.4
- werkzeug: 3.0.1
- wheel: 0.41.3
- widgetsnbextension: 3.6.6
- wordcloud: 1.9.2
- wrapt: 1.14.1
- xarray: 2023.7.0
- xarray-einstats: 0.6.0
- xgboost: 2.0.1
- xlrd: 2.0.1
- xxhash: 3.4.1
- xyzservices: 2023.10.1
- yarl: 1.9.2
- yellowbrick: 1.5
- yfinance: 0.2.31
- zict: 3.0.0
- zipp: 3.17.0
* System:
- OS: Linux
- architecture:
- 64bit
- ELF
- processor: x86_64
- python: 3.10.12
- release: 5.15.120+
- version: #1 SMP Wed Aug 30 11:19:59 UTC 2023

### More info

_No response_

cc @borda @carmocca @justusschock @awaelchli

Start with the linked Colab reproduction and trace how a LightningModule's registered buffers are handled during Trainer setup and precision conversion. The fix is done when an explicitly float64 registered buffer remains float64 after setup under a different Trainer precision; the issue names no repository file or regression test.

Written by the indexing model from the issue text.