No Registry, just cards

I don't think you need to say anything at all. Since you don't have a registry, guests are going to default to cash gifts, and they're going to give you a card with that gift. 95% of our guests gave us cash, check, or gift cards and not one of them came without a card.

I would change the wording a little bit to make it clear that you're looking for sweet notes from your guests, not cards filled with cash.

I think what you said is perfect and I wouldn't change it. I think putting something helps guests and you made it clear that the cash isn't important but the cards for a scrapbook. Well done!

I wouldn't write anything. If you're not registering, people will give cards with cash or checks most likely. Even if you were registering, most people send a card as well. It just seems very disingenuous to me to say essentially "we definitely don't want anything, but if you want to give us something - this is what we want." Just dont' say anything and you will get tons of cards anyway.

I wouldn't even mention it in my opinion. I think your guests will see on your invitations/website that you aren't registered anywhere. and a majority of your family and friends will already know you moved and had a house warming.

my DH and i didn't register, had no engagement party nor bridal shower. we didn't mention anything but put out a box and received tons of cards.

I agree with PPs that not mentioning it would be better, in my opinion! Most people will bring cards anyway

Another vote for saying nothing. People will figure out you don't have a registry and they will very likely bring/send cards -- typically with cash or a check. Saying anything, without getting awkwardly specific about what you do and don't want, is easy to misinterpret.