[Task] Extend cypress tests to use real browsers

[eike-hass]

Description

Use headless Chrome and/or Firefox instead of electron as a target for tests in CI.

Motivation

We missed an issue in #1386 that was not caught by testing in electron.

Change checklist

Add an x to the boxes that are relevant to your changes, and delete any items that are not.

• The feature or fix is implemented in Rust and across all bindings whereas possible.
• The feature or fix has sufficient testing coverage
• All tests and examples build and run locally as expected
• Every piece of code has been document according to the documentation guidelines.
• If conceptual documentation (mdbook) and examples highlighting the feature exist, they are properly updated.
• If the feature is not currently documented, a documentation task Issue has been opened to address this.