The homepage logging you back seems to be intentional. Its part of the new web system that's slowly coming out. Account sessions are managed through the Jagex website now, and then product websites can authenticate through that to generate your session to use whatever site you need to use at that time (this case, the RS site).
You still have to relog if you want to access any higher privileged/sensitive pages like: account settings, payments, solomons, etc. You also would need to re-auth to access the Jagex Account pages.
You'll know when you're logged out by actually logging out of the website when you're finished using it. And going with what Miles said, this is standard advice for anything. Many other websites offer prolonged login sessions on their websites. If users follow basic guidelines, there's not a huge security risk....else they'd all be designed a bit differently.
Can I turn in a paper without citing all sources?
"No."[1]
1. William Shakespeare,
Hamlet
, Act III, Scene 1, line 96.
20-May-2023 00:58:35