In my opinion,,,
していて has some different meanings actually, but don't think it's difficult or hassle coz it's a convenient word!
the way he taught you is right also ; )
another meaning
していて( is doing) is coming from する(do)
昨日パーティ行けなくてごめんね!宿題をしていて行けなかった
I'm sorry that I could't go to the party yesterday! I was doing homework
していて change the form sometimes
→してて(casual way of していて)
→していた(was doing)
→してた(casual way of していた)
Gosh it's pretty hard to explain the first language lol
Let me know if you have any other questions at anytime ;)