I have an awsome breed for you!! It is called a stuffed dog and I would suggest the Gund brand or the TY brand. They withstand the loving affection of a 1-2 year old, they don't jump, they don't bite, and they don't bark! What more could you want from a dog?
No offense, but a dog is going to jump, bite, nip, pee on the floor, and wreck havoc in your house. I was shocked that you even considered a Border Collie with children so young and that you got rid of it just because it jumped on your little boy. I'm still in shock. Yes, I can understand that it injured him, but when Rose used to jump on us and we'd have nice long bleeding rake type scratches down our backs we didn't get rid of her. We just trained her not to jump. (Isn't that a strange concept...training a dog. Huh. Who would have thought?)
Ok, now that I got that out of my system and that I have a very lousy first impression, here is my advice:
1.) I really do suggest you get a stuffed dog, just like I said earlier. They are awsome family pets.
2.) If you are really set on getting a dog, I'd wait. You're children are too young for a small dog, and apparentally, you don't want to take the time to train a larger dog to stop them from injuring your kids, so a large dog is out of the option too.
You should not buy a dog for your child. That is just a no no. Especially one so little. My 10yr old sister still doesn't have enough responsibility to handle a dog, what the heck makes you think a one your old can?
If you do decide to get a dog, get an adult dog that is already trained, housebroken, and is lazy enough to not jump on anybody or knock your kids down. I saw that somebody mentioned a Lab. Labs are wonderful dogs and I find it terrible that you are judging the whole intire Lab breed on one experience. How sad. Anyways, if you do look into one, do NOT NOT NOT NOT NOT NOT NOT NOT NOT NOT get one under two years of age, becasue they will jump and chew like no other. My five year old sister is terrified of Sadie (or Lab pup) because she jumps and nips. I shudder to think how she would be with a baby.
Also: a toy breed is our of the question. They are just as capable as hurting a little child as a big breed and they are so fragile that a little child could kill them easily.
Think through this very carefully before you get another dog and then find out it is dead because your toddler squished it, or it too jumps and nips on your kids so it ends up in another home because you didn't do your research.