HTML Architect: Improve the stability of the e2e Playwright tests

The “deployment version creation” tests sometimes fail in the pipeline

Also, the “show-sql” test doesn’t pass on the first try in CI/CD

The same problem was in vim e2e test

And locally, “Entity Relationship creation with incremental search” sometimes doesn’t work.
This needs to be fixed so it doesn’t slow down team for pushing code to the repository

We should also double-check that the tests have no other issues that they are healthy, stable, and don’t produce any unnecessary warnings