Lil B - while I did read your post in its entirety, your puppy is telling you that he wants
to toilet outside, like a normal dog should. I would not be fooled by his size, or the climate. I live in London, Ontario (Great Lakes Basin) and we have the same cold weather you have.
Oliver came home at 8 weeks in February and learned quickly how to go outside and come back inside. Quite frankly, I think the colder weather is helpful as they don't start messing about and think it's playtime, plus the cold induces them to go more quickly.
Your puppy has a coat that nature gave him. I think that although it would be easier on you humans to not have to brave the elements in the Winter, your puppy will do just fine, and be in and out in a matter of minutes.
It is confusing to the pup, especially when he does not like the puppy pads, to make him try to go in a place that feels unnatural. He will learn that the grass in under the snow. I don't think you mentioned if you were in an apartment or not as that does make a difference. Since you did say "house", I am going to presume that you are able to access the outdoors with one door, as opposed to a long elevator wait and ride. By the time it is really cold, your puppy will be able to handle the short time outdoors.
Oliver was very tiny when he came home, (under 5 pounds) with a very thin coat, unlike your pup. He did not become ill, or hurt by going outside to do his business. It was actually easier housetraining him than it was Henry who came home in June to the nice weather.
We made a barrier of sorts out of the snow, which the puppy was unable to climb, nor had the desire, and all we had to do was set him outside the door (same one every time) and he would go instantly. By the time Spring came, he was looking for snow on which to do his business.
Kian suggested those Potty Patches, and other people actually litter train their dogs like a Cat, but I always thought those were mostly for people in highrise buildings who weren't able to quickly put their dog out.
I think it is counter productive to try and turn back the training as that is confusing to your puppy. Just because he is small, he is no less of a dog. JRT's for example have very thin, short coats (for the most part) and they manage in all sorts of weather as puppies. By the time the snow flies, your pup will be well past 4 months of age and should do just fine in the cold weather for the short amount of time it takes to do its business, IMHO. Ultimately it is less work for you.