It seems as though the bond between the hydrogen and the double bonded carbon is acting as a nucelophile, but I know that cannot be correct.
You're actually on the right track. Your scheme for that step is written a little confusing. What happens when you react t-butyllithium with the vinyl ether?
I would expect it to attack the double bond, like an anionic polymerization initiator, but it seems kind of bulky for that, and I am pretty sure vinyl ethers can not be polymerized anionically.
also, it seems as though i you gave it enough time you would form ethylene and vinyl lithium oxide. but this doesn't make sense here.
I was taught that vinyl hydrogens are not acidic at all, but based on this reaction and the strength of the base it seems like that might be happening. I feel strange saying this though.