Pauli exclusion principle and Entangled pairs

Let's start from the definition of entangled state.

Briefly -- if the state of your system can be described by separately defining the states of its components, then we call the state of this system a separable state.

If such a description is impossible -- then the state is an entangled state.

Now, for both your examples it is impossible to factorize the states of individual particles in the description of the state of the total system. Therefore both of these states are entangled states.

Every state can be written in the way you mention, in terms of two states 1 and 2, for an appropriate choice of states 1 and 2. By itself it does not indicate any entanglement. What makes a state entangled is a specific property of the two states 1 and 2, namely that they are physical states belonging to two subsystems which do not interact with each other (for example systems spatially separated from each other). Only then it is interesting to talk about entanglement, which is roughly speaking the degree of correlation between the two states, which cannot be undone by operating on either one of the two subsystems separately.