For proof of what you are explaining, the "original WiP thread ..." is looking sufficient; I had a question here, whether you needed to actually have submitted the game at the other jam so I checked the wording and it says "[...] finish a game you started or remade at another Jam this summer" but also, in the rules "must have been entered into another jam".
So I think strictly speaking you should have submitted your game at the other jam first (did you?), but I'll be putting the emphasis on quality here not nit-picking; so as long as you could demonstrate (as explained by you) that you participated in that jam your entry will qualify.