I/O roundtrip for annotations from a cropped raw induces a shift of raw.first_samp samples

If you crop a raw (with no measurement date), inducing a change in raw.first_samp, then save and reload the annotations attached, they are shifted by raw.first_samp. Let's consider a raw recording which has never been cropped and starts at raw.first_samp = 0, with 2 annotations at 2 and 4 seconds lasting 0.5 seconds:

from mne import Annotations, read_annotations
from mne.datasets import sample
from mne.io import read_raw_fif


fname = sample.data_path() / "MEG" / "sample" / "sample_audvis_raw.fif"
raw = read_raw_fif(fname, preload=False)
raw.pick_types(eeg=True)
raw.crop(0, 10, include_tmax=True)
raw.load_data()
raw.set_meas_date(None)  # remove measurement date since it impacts annotations
raw.set_annotations(None)
raw._cropped_samp = 0  # overwrite first_samp to 0 for this example
annots = Annotations(onset=[2, 4], duration=0.5, description="test", orig_time=None)
raw.set_annotations(annots)
raw.plot(n_channels=5)

Now let's crop the first second and set raw.first_samp to the 601 sample.

raw.crop(1, 8, include_tmax=True)
raw.plot(n_channels=5)

And finally, let's save the annotations to a separate file and load them back:

fname = "~/Downloads/test_annot.fif"
raw.annotations.save(fname, overwrite=True)
annots = read_annotations(fname)
raw.set_annotations(annots)  # overwrite existing annotations
raw.plot(n_channels=5)

By the end of the I/O roundtrip, the annotations are shifted by raw.first_samp, in this case 1 second. The "fix" is to edit the .onset attribute:

fname = "~/Downloads/test_annot.fif"
raw.annotations.save(fname, overwrite=True)
annots = read_annotations(fname)
annots.onset -= raw.first_samp / raw.info["sfreq"]
raw.set_annotations(annots)  # overwrite existing annotations
raw.plot(n_channels=5)

But Annotation.onset is not documented, I don't expect users to edit this attribute.

In the Notes of the Annotations class, there is this warning:

This means that when raw.info['meas_date'] is None, doing raw.set_annotations(raw.annotations) will not alter raw if and only if raw.first_samp == 0. When it’s non-zero, raw.set_annotations will assume that the “new” annotations refer to the original data (with first_samp==0), and will be re-referenced to the new time offset!

which seems related and is completely unclear to me.

raw.set_annotations(raw.annotations) will not alter raw [...]

Alter raw? How would that alter it if the raw.first_samp is different from 0?

Anyway, IMO, this I/O roundtrip should just work without users requiring to edit undocumented attributes of the re-loaded Annotations.

Start with the reproduced sequence using raw.crop, raw.set_annotations, Annotations.save, and read_annotations, then inspect how first_samp and meas_date are handled during annotation assignment and I/O. Verify the fix by rerunning the example and confirming that reloaded annotations retain their positions without manually changing onset.