[big]

This is probably a really trivial question, but if the problem says that the cathode chamber is 500 mL during the electrolysis of water, does that mean that this chamber only includes the solution and that the gas goes in some different chamber? Or does "cathode chamber" typically include both the gas and the solution?

I googled some pictures for the electrolysis of water, and while some would seem to imply that the word "cathode chamber" includes both gas and solution, others seem to imply that the gas bubbles out of the solution and no longer counts as part of the chamber, so I wasn't sure.

[Arkcon]

Its impossible to tell.  Like you've said, you've found pictures where it can be both.  So without more context of the stated problem, we really can't say for sure.