Trigger CI tests after bumping the version and before publishing

[carmenberndt]

We want to run e2e tests for each commit, even for the ones from the CI.

Currently checking for updates, bumping the versions and publishing the extension happen in a single step in a job in a Github Action workflow. (for jobs publish and publish-dev)
(Note: for stable, bumping and publishing happens in memory (most changes are not committed to the repo)

Bumping the versions should be a separate step in the job, triggering the e2e tests and then publishing should happen via another step in the job.

[carmenberndt]

Next steps:
Write down the full solution first, to the exact step as a spec before implementing this.

[carmenberndt]

Step 1: check-update
This returns a boolean output describing if new updates are available or not.
Step 2: Bump versions
This only runs in case the output of check-update is true.
Changes are only committed for the unstable version.
Step 3: Run headless tests
This only runs in case the output of check-update is true.
Step 4: Publish Extension
This only runs in case the output of check-update is true.