YES if you claimed on their or your insurance.
but since the other driver was found to be at fault, then put it down as
"not our fault" or something similer.

Yes. WHen you select Yes, there'll be another option where you can put
"Costs Recovered from 3rd Party" or similar.

Are you genuinely thick?
Read the policy or listen to what the person asks you on the phone.  Give
the answer to the question, not the answer to a question you would like to
be asked.  If you lie or say you didn't understand something you have signed
to say you did, then you're stuffed.

IME almost any communication with your insurance company is counted as a
"claim" by them and therefore a method by which they can put your premium up. ;-)

In actuality, any event that involved insurance no matter who was at fault is
a "claim". They'll want the details of the claim and will record it as a "no
fault" claim. Even so they'll still raise your premium accordingly as you're
the sort of driver that gets driven into.

Remember, insurers are interested in risk: the possibility of having to pay
money out, or even having to administer an incident. They charge you according
to how much they feel they're likely to have to fork out any of the delicious
lolly they fleeced you for. If you have any accidents then obviously you're a
higher risk. They don't give a damn about fault or whether you could've
avoided it, they just care about paying out, or even the thought of it.