Screen Record the browser

Check out Screenium if it suits your needs, I have used it for a few projects and it worked great.

Here in 2020 - macOS Catalina and later allows the built-in /usr/sbin/screencapture tool has options for specifying the capture area. And if you use it graphically, it remembers the last selection so unless you change it, you can relaunch the app and get the same settings without the details below.

  -R<x,y,w,h> capture screen rect
  -v        capture video recording of the screen

For example:

screencapture -R10,10,500,500 -v cap.mov

There's a bunch of other good stuff in the man page:

man screencapture