I'm replying to my own thread here in case anyone else is searching for the answer to this. I feel rather silly, but still thought it would be best to share the konwledge:
You get the TestFlight code by clicking on the "Start Testing" button in the email, even if you're reading the email on a different device than the device with TestFlight installed. Clicking the button opens up a new webpage (at least that's what it did on my desktop) with simple instructions on installing TestFlight, including the Redeem code.
So now I know - next time I should be slightly more adventurous I guess....
Yes, this fits my experience & testing.
I've had some testers who had a hard time because their corporate email systems did not like the Start Testing button in the email. Mostly I solved this by getting an alternative email address from them.
But I also found that when you do get a redeem code by clicking on Start Testing, the code has to be used in a short time (or it says Expired), and that it cannot be used twice. Also, if you've already installed the app, you cannot get a code.