Tim:
I would try erasing *just* the cache. But, first, try a forced reload. It should re-fetch the page, rather than just refresh the display.
Michael Hennebry:
I tried reloading a number of ways. They all seem to have "worked". Noe were effective. I tried both reloading the page I clicked on and the page I got to. No go. Firefox does not seem to offer the option to delete just the cache. It did allow picking and choosing websites. I picked php8.ltd because that was the site for both pages. No go.
A problem with clearing just a particular site is that your problem site may be a combination of more than one service, with other things in the background that aren't readily obvious, and one of them was the cause.
If you have an add-on like NoScript, you might want to check you haven't accidentally forbidden something you need. Alternatively, an add-on like that does show how a single-site drags in content from a plethora of sources, and might just help you find the problem one.
Also, double-check on the problem system that you don't have a typo in the address you're trying to load. And, on that note, if there's an alternative way to log into the site (through a different entry page), see if that works.
Failing that, elaborating on my earlier suggestion:
In Firefox's setting, go into the privacy & security section, there's a cookies and site data sub-section with a clear button. That "clear" button pops up a window with options, rather than simply clears everything when you hit it. Depending on your version you may have options like
Cookies and site data Cached web content
or
When (last hour & various other time-based options) History Cookies and site data Temporary cached files and pages Site settings
You don't have to lose your history, at all. Unless there's something seriously wrong with Firefox, I don't see any way that could be a part of a the problem. It's just a trace of where you've been.
You should be able to just purge the cached files and nothing else. Generally, that's all that's required if something has got stuck in the cache and is confusing things. And it shouldn't cause another problem, in itself, the browser will just refetch what it needs when you visit a site again.
You might have to clear cookies, which would be more a site's problem than your browser problem, but I'm yet to have to do that (for this kind of thing). Of course that almost certainly means anything that you automatically log into without having to type your password again, will require you to do that. But the chances are that the auto-fill in feature will supply the names and passes with you just having to start typing in your logon name.
The site settings thing is a new one on me. I think it's just when people have customised their browser for a site, rather than customised a site through their browser. e.g. You've picked dark mode, or changed font sizing.
There's a separate preferences for deleting cookies on exit, with a pop-up window for exceptions. You could look into that.