This is something worth paying attention to. Aside from the highschool/college exam flashback anxiety you talk about, the Hawthorne Effect suggests that "individuals modify or improve an aspect of their behavior in response to their awareness of being observed", and some might argue that it's practically impossible to to "observe" a natural use of a product or website (without resorting to secret recording). That said, there's still at least some value in testing users, so we should probably strive to mitigate this effect with language.
In the approach, you could ask people to take part in a "Website Evaluation", rather than a "Usability Test". You could play around with other works like "assessment" and "analysis", always making sure that the website or app is the subject of the sentence.
When it comes time to carry out the "website evaluation", read them Steve Krug's Usabilty Script, part of which reads:
We're testing a web site that we're working on to see what it's like for actual people to use it.
I want to make it clear right away that we're testing the site, not you. You can't do anything wrong here. In fact, this is probably the one place today where you don't have to worry about making mistakes.
We want to hear exactly what you think, so please don't worry that you're going to hurt our feelings. We want to improve it, so we need to know honestly what you think.