Enable multiple loggers to save to the same version directory

Bug description

As the title said, Tensorboard and CSV loggers create different directories when being used together. They should be in same version folder.

What version are you seeing the problem on?

v2.0

How to reproduce the bug

from typing import Any, Union

import lightning.pytorch as pl
import torch
from lightning.pytorch.callbacks import (
    LearningRateMonitor,
    ModelCheckpoint,
    RichProgressBar,
)
from lightning.pytorch.demos import Transformer, WikiText2
from lightning.pytorch.loggers import CSVLogger, TensorBoardLogger
from lightning.pytorch.utilities.types import STEP_OUTPUT
from torch.optim import AdamW
from torch.utils.data import DataLoader
from transformers.optimization import get_cosine_schedule_with_warmup


class BoringTransformer(pl.LightningModule):
    def __init__(
        self, vocab_size: int, learning_rate: float = 5e-5, warmup_ratio: float = 0.1
    ) -> None:
        super().__init__()
        self.learning_rate = learning_rate
        self.warmup_ratio = warmup_ratio
        self.transformer = Transformer(vocab_size=vocab_size)

    def training_step(self, batch) -> STEP_OUTPUT:
        x, y = batch
        z = self.transformer(x, y)
        loss = torch.nn.functional.nll_loss(z, y.view(-1))
        self.log("train_loss", loss)
        return loss

    def configure_optimizers(self) -> Any:
        opt = AdamW(self.parameters(), self.learning_rate)
        opt_list = [opt]
        lrs = {
            "scheduler": get_cosine_schedule_with_warmup(
                opt,
                self.trainer.estimated_stepping_batches * self.warmup_ratio,
                self.trainer.estimated_stepping_batches,
            ),
            "interval": "step",
            "frequency": 1,
        }
        lrs_list = [lrs]
        return opt_list, lrs_list

pl.seed_everything(1312)
dataset = WikiText2()
dataloader = DataLoader(dataset, batch_size=64)
transformer = BoringTransformer(dataset.vocab_size)
trainer = pl.Trainer(
    enable_checkpointing=True,
    default_root_dir="boring_logs",
    accelerator="gpu",
    devices=1,
    precision="16-mixed",
    logger=[CSVLogger("boring_logs"), TensorBoardLogger("boring_logs")],
    callbacks=[
        RichProgressBar(),
        ModelCheckpoint(every_n_train_steps=16, save_last=True)
    ],
    max_epochs=1,
    accumulate_grad_batches=8,
    log_every_n_steps=16,
    enable_model_summary=False,
)

ckpt_path: Union[
    str, None
] = None
trainer.fit(transformer, dataloader, ckpt_path=ckpt_path)

Error messages and logs

boring_logs
└── lightning_logs
    ├── version_0
    │   ├── checkpoints
    │   │   ├── epoch=0-step=112.ckpt
    │   │   └── last.ckpt
    │   ├── hparams.yaml
    │   └── metrics.csv
    └── version_1
        ├── events.out.tfevents.1689579440.939aa81ccfa0.28.0
        └── hparams.yaml

4 directories, 6 files

Environment

More info

Kaggle notebook: https://www.kaggle.com/inhanhv/tensorboard-and-csv-in-same-directory

cc @borda @awaelchli @Blaizzy

Reproduce the issue with the provided Trainer configuration using CSVLogger and TensorBoardLogger, then trace how each logger selects its version directory. Confirm that both loggers write into the same version folder and add or update coverage for the combined-logger case.