Is it possible to get the query to work with both forms without showing the parameter pop up for the closed form? ...Or do I have to create 2 identical queries?
What I think you're asking for is doable but requires coding. You cannot expect a query to simply default to one of the OR criteria statements simply because it cannot resolve the other one. It's like saying, If I say to you "Do A if balloon is blue or do B if ball is black" and I don't show you the ball, what are you going to say? Show me the ball...
You might be able to call a function that checks to see which form is open and then returns the value from the correct control on the opened form.
The more we hear silence, the more we begin to think about our value in this universe.
Paraphrase of Professor Brian Cox.