I'm going to stick with Gmail as my example. It would be helpful, if it's happening on a site that does not require login, if you would give the URL where you're having problems.
If I am in the Gmail inbox, if I fire up NVDA Find, the announcement from NVDA is, "Find dialog, type in text you wish to find," while if I fire up the browser find NVDA says, "Find in page inbox, " followed by more verbiage. If I'm in the Sent box, then substitute "sent" for "inbox." The announcement for the NVDA find is invariant.
As far as I can tell, and this is what I'd expect, the NVDA find is looking for the text you give it, with or without case sensitivity depending on your checkbox state, on the page being displayed in the browser. The same is true of the web browser search itself. The same is not true if you use the Search Mail box that's part of the Gmail page, which will search all e-mail messages in sender, subject, and message bodies for each message.
Without literally seeing what's happening, or having some way to know definitively what you're doing via other means, I can't say much more.
Brian - Windows 10 Pro, 64-Bit, Version 1903, Build 18362
The color of truth is grey.
~ André Gide