I’m certain that my Hotmail emails stay on the Hotmail servers even when I have downloaded them via Applemail. If I delete in AppleMail then they also delete from Hotmail. But even then Hotmail (web) can allow me to recover deleted emails. When I tried it once some 4000 ish emails appeared for me to sift through.Outlook, Thunderbird and most email clients run locally...they are not cloud based. Depending on the email protocol you use - POP or IMAP - your emails will be downloaded to your own machine in the case of POP, or left on the cloud server in the case of IMAP.
It's true though that even with POP the emails will reside in the cloud until you download them by running your email client. But that's not the same as cloud based email services.
(There may be an exception in some POP protocols where you can opt to not delete the emails after you've downloaded them but I don't see the point of that)
Even if you have two different machines with your email clients set to POP on each, you will split emails between whatever machine your currently running.
Confusing, or what?