differences - American English: which vs that

We've had an American Americanise some phrases for us (with the point of teaching children English), e.g:

UK English: There’s a little white dog which lives on the second floor.
US English: There’s a little white dog that lives on the second floor.

But there are three sentences which were not altered. We would rather not have a mixture of that and which to make things less confusing for the students. The sentences that were not changed were:

I really like this pencil case which Rose gave me.
There’s a store near here which is open all night.
This is a really cool picture which I brought with me from Little Bridge.

Should these be changed to that, or should we change the other sentences that have already been translated back to which?

Answer

I am not a native speaker but at school I was taught that you should use "that" for defining relative clauses, whereas both "which" and "that" are allowed for non-defining
relative clauses.

Therefore:
"I really like this pencil case which Rose gave me."
Here, the relative clause only adds information about a pencil case that is already known
(this pencil case). In this case, as far as I know, you can also use "that".
The relative pronoun cannot be omitted.

"I really like the pencil case that Rose gave me."
Here the relative clause defines the pencil case: I am talking about the pencil case
that Rose gave me, not about another one. In this case, it is compulsory to use "that".
Alternatively, you can omit "that" altogether:

"I really like the pencil case Rose gave me."

This is at least the rules that I recall from school and I may be wrong.
I also do not know if the American and the British use differ on this since I am neither
from Great Britain nor from the United States.