According to both websites User Testing is basically observing real people test the functionality of a website, app, or other digital product.
The definitions of these are pretty straight forward. With moderated and unmoderated your either supervised or not. With In-person and remote you're either on site with the tester or not.
User testing is important to understad how your site works, if there are any bugs or errors, to confirm your website meets client expectations, and getting feedback for potential changes
Another way User Testing is done is through surveys. These are good if you need a large range of responses in little time. Since you can choose when and where you do these tests it makes it flexible for your needs and for others.
From my understanding to test a website on different versions of browsers the tester would need to install multiple browsers and even operating systems on different devices such as a desktop, laptop, or mobile device. Then run your website on all of these. Obviously this would help determine things such as your website functioning properly across multiple devices. Maybe if people prefer a specific browser or device.