I have my games embedded on my own website. The website is used in the computer lab where I teach. My students play my (educational) games all the time but the traffic is never reflected in analytics. It seems like traffic is only tracked if a person plays a game directly from the itch.io website.

Which analytics? 

I can see how many people play my browser games on my website through Awstats. 

Awstats shows how many people accessed a certain file which is needed to run the game (under Page URL - Top 25). That stat equals how many people started to play the game. 

I can also see under the heading Top Downloads, how many times the sound files (as an example) were downloaded (it's automatically downloaded once they open the game). 

Awstats is normally included if your website hosting includes a control panel. Find it under "metrics"