If the Fitbit is recording that movement, then yes, you would be double-counting. But if it's not (like when you mentioned the upper body strength training that it's not picking up), then you could manually log it and you wouldn't be double-counting.
Does that help?
"You may have a fresh start any moment you choose, for this thing that we call "failure" is not the falling down but the staying down." Mary Pickford
"No matter how slow you go, you are still lapping everybody on the couch."