My guess is that those terminal programs are designed to run in your browser and AFAIK for security reasons browsers generally don't have access to the PC's file system.
This webpage that I found explains how to install PuTTY on a Chromebook. I don't have a Chromebook so I have no idea whether it works or whether it does what you want on your Chromebook but the regular PuTTY does allow you to save the incoming data to a file.
...R