It sounds like a fairly 'standard' proxy issue, but those can vary a lot in configuration.
ZAP is proxying other traffic as normal, I assume?
Since it looks like you're using windows, I recommend using Fiddler as a proxy, which will seamlessly integrate itself with whatever your internal network settings are.
Then, point ZAP to that (usually localhost:8888), then point your browser to ZAP.
Then run your little experiments as before, and you can compare the GET request as sent from the browser directly to fiddler, and the one from ZAP to fiddler and see what/if there are any differences.