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That's not what a collision is, a collision is two separate generators generating the same private key that points to the same address. Not two different keys pointing to the same address. The private key is the root of an address. The address is simply the hash of that private key.
address is not hash of private key, it is hash of public key. and what you explained (2 different entities find the same address with 2 different keys) is only one type of collision. the other type is indeed 2 different entities finding the same private key. the space they are in are different. the first one is in 160 bit and second is in 256 bit and both of them are huge enough to be considered impossible.
That is NOT what I explained, two different private keys will never link to a single address. But two different scripts can and have on several occasions generated identical private keys that have been tied to already used, and in some cases wallets with balances.
Secondly, It's a hash that roots from the private key, same difference.