Hey Doone! Congrats on linking to the cloud by the way! Check out the risk insights, that's my all-time favorite part.
I'm not official support, and the Famous Esben is on holiday - but the patch status is in the on-prem report currently - maybe hit up the page and re-copy it. (reference https://www.lansweeper.com/resources/report/patch-tuesday/microsoft-august-2024-patch-tuesday-audit/)
Regarding the report differences - Lansweeper Sites (a.k.a. cloud) reporting has it's share of advantages and limitations... the LS Sites reports and custom views are easier to build versus knowing SQL, however it is more limited in functionality/logic at this current point in time.
To answer your last question - personally as a sysadmin, I use on-prem more to create more custom/granular reporting that requires computation/ranking/etc for my teams to use. I use LS Sites reporting and views for other general IT audiences to use as it is easier to build simpler reports, and when I want to lock down what folks can and can't see. So, yeah I use the on-prem patch tuesday report because I end up making five or so modified/custom reports from it š