Refactor Rest Binary sensor with ManualTriggerEntity

[gjohansson-ST]

Proposed change

SSIA

Requires: #97398

Type of change

• Dependency upgrade
• Bugfix (non-breaking change which fixes an issue)
• New integration (thank you!)
• New feature (which adds functionality to an existing integration)
• Deprecation (breaking change to happen in the future)
• Breaking change (fix/feature causing existing functionality to break)
• Code quality improvements to existing code or addition of tests

Additional information

• This PR fixes or closes issue: fixes #
• This PR is related to issue:
• Link to documentation pull request: Rest Binary Sensor to ManualTriggerEntity home-assistant.io#28526

Checklist

• The code change is tested and works locally.
• Local tests pass. Your PR cannot be merged unless tests pass
• There is no commented out code in this PR.
• I have followed the development checklist
• I have followed the perfect PR recommendations
• The code has been formatted using Black (black --fast homeassistant tests)
• Tests have been added to verify that the new code works.

If user exposed functionality or configuration variables are added/changed:

• Documentation added/updated for www.home-assistant.io

If the code communicates with devices, web services, or third-party tools:

• The manifest file has all fields filled out correctly.
  Updated and included derived files by running: python3 -m script.hassfest.
• New or updated dependencies have been added to requirements_all.txt.
  Updated by running python3 -m script.gen_requirements_all.
• For the updated dependencies - a link to the changelog, or at minimum a diff between library versions is added to the PR description.
• Untested files have been added to .coveragerc.

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[home-assistant]

Hey there @epenet, mind taking a look at this pull request as it has been labeled with an integration (rest) you are listed as a code owner for? Thanks!

Code owner commands

Code owners of rest can trigger bot actions by commenting:

• @home-assistant close Closes the pull request.
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• @home-assistant reopen Reopen the pull request.
• @home-assistant unassign rest Removes the current integration label and assignees on the pull request, add the integration domain after the command.

[emontnemery]

Same comments as on the switch PR.

[home-assistant]

Please take a look at the requested changes, and use the Ready for review button when you are done, thanks 👍

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[emontnemery]

Nice! Thanks @gjohansson-ST 👍

[gjohansson-ST]

Coverage not affected

---------- coverage: platform linux, python 3.11.4-final-0 -----------
Name                                             Stmts   Miss  Cover   Missing
------------------------------------------------------------------------------
homeassistant/components/rest/__init__.py          109      1    99%   198
homeassistant/components/rest/binary_sensor.py      71      0   100%
homeassistant/components/rest/const.py              20      0   100%
homeassistant/components/rest/data.py               73      5    93%   115-119
homeassistant/components/rest/entity.py             39      0   100%
homeassistant/components/rest/schema.py             14      0   100%
homeassistant/components/rest/sensor.py             74      0   100%
------------------------------------------------------------------------------
TOTAL                                              400      6    98%