BUG: DataFrame.to_string() adds a backslash at the end of the wrong line in Pandas 2.0.1

[GianlucaFicarelli]

Pandas version checks

• I have checked that this issue has not already been reported.

• I have confirmed this bug exists on the latest version of pandas.

• I have confirmed this bug exists on the main branch of pandas.

Reproducible Example

import pandas as pd
df = pd.DataFrame({"x": [1, 2, 3], "y": [4, 5, 6]})
print(df.to_string(line_width=1).replace(" ", "."))

Output (Pandas 2.0.1):

...x...
0..1..\
1..2...
2..3...

...y..
0..4..
1..5..
2..6..

Output (Pandas 1.5.3):

...x..\
0..1...
1..2...
2..3...

...y..
0..4..
1..5..
2..6..

Issue Description

When df.to_string() doesn't fit the line width, in Pandas 1.5.3 a backslash is added at the end of the headers line, or at the end of the first data line if no headers are present.

In Pandas 2.0.1, instead, the backslash is added at the end of the first data line even when the headers are present.

The regression is a consequence of the changes in #49738

I think that the old behaviour is desired, and I can submit a PR with a possible fix.

Expected Behavior

The backslash should be printed at the end of the headers line when they are present, as it was in Pandas 1.5.3.

Installed Versions

INSTALLED VERSIONS

commit : 37ea63d
python : 3.10.8.final.0
python-bits : 64
OS : Linux
OS-release : 3.10.0-1160.42.2.el7.x86_64
Version : #1 SMP Tue Aug 31 20:15:00 UTC 2021
machine : x86_64
processor : x86_64
byteorder : little
LC_ALL : en_US.UTF-8
LANG : en_US.UTF-8
LOCALE : en_US.UTF-8

pandas : 2.0.1
numpy : 1.24.2
pytz : 2023.3
dateutil : 2.8.2
setuptools : 67.7.2
pip : 23.1.2
Cython : None
pytest : None
hypothesis : None
sphinx : None
blosc : None
feather : None
xlsxwriter : None
lxml.etree : None
html5lib : None
pymysql : None
psycopg2 : None
jinja2 : None
IPython : 8.12.0
pandas_datareader: None
bs4 : None
bottleneck : None
brotli : None
fastparquet : None
fsspec : None
gcsfs : None
matplotlib : None
numba : None
numexpr : None
odfpy : None
openpyxl : None
pandas_gbq : None
pyarrow : None
pyreadstat : None
pyxlsb : None
s3fs : None
scipy : None
snappy : None
sqlalchemy : None
tables : None
tabulate : None
xarray : None
xlrd : None
zstandard : None
tzdata : 2023.3
qtpy : None
pyqt5 : None

[GianlucaFicarelli]

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