The usual case to distinguish a pseudorandom function from a random function is to assume that the adversary can choose the plaintext blocks. Is there another case (game) in which the adversary can not choose the plaintext blocks ?
There are various adversary models, in fact it is typical to test our schemes against multiple adversaries to prove various nuances of security.
The most intuitive of all is an adversary that can produce the plaintext (or a part of it) given only the ciphertext. An extension to this model, stronger than the other, is the one you mentioned, letting the adversary select a number of plaintexts $n$ (which could be infinite), give him the ciphertexts and some random data and let him decide which is which.
In general we require the adversary to produce a result in polynomial time because while many of the primitives can in theory be broken, the probability of this happening in polynomial time is considered negligible.
These are called weakly pseudorandom function families.