[CI] Use custom image and improve caching for Cypress

Hah, @larowlan already opened an upstream issue for this: ✨ Would you accept a Cypress container Active .

But @fjgarlin at #3520918-3: Would you accept a Cypress container → told us we can create or own, which this issue is about 👍

Ugh - it seems to work, up until the point all the Cypress tests fail in the setup step and the output and screenshot is entirely unhelpful.

Yeah, having an official image would only make sense if cypress will be used for core too.

Custom image seems better rather than official one imo, unless we are planning to include cypress testing in core.