Update gha that publishes website

[ElsLommelen]

Description

This PR replaces an old external GHA by own code for retrieving the branch name of a recently merged branch to avoid the deprecation notes described in issue #111.

Task list

Steps by contributor:

• Add description to this pull request (under "## Description")
• Mark the pull request as 'ready for review'

Review steps for the author(s):

• Add reviewers, at least one subject-matter specialist and one administrator
• Wait for review comments and address them
• Iterate until reviewer approvals (merging the pull request will be done by an administrator)
• Verify that the checks done by continuous integration succeeded. These will check if protocolhelper::check_frontmatter() and protocolhelper::check_structure() succeeded without errors.

To be done by an administrator after review: see guidelines for admins.