The way I was introduced to wavefunctions was in form of Dirac notation: $$\psi(x)=\langle x| \psi \rangle$$ i.e. the probability amplitude of going from state $\lvert \psi \rangle$ to state $\lvert x \rangle$, where state $\lvert x \rangle$ is a member of eigenvectors of some observable, forming a basis. It also makes intuitive sense because the wavefunction peaks at a specific $x$ values which happens to be eigenvalues of the state $\lvert \psi \rangle$. Another example of wavefunction as the probability amplitude is in this notation: $$\lvert \psi \rangle = \sum_i \psi(\textbf{i}) \lvert i \rangle $$ (eigenvector basis expansion of an arbitrary state).
But now, I am being exposed to notation of the sort $X \psi(x)=x \psi(x)$ or $A \psi = \lambda \psi$, which is rather confusing because the wavefunction is being treated like a ket vector or state rather than a probability amplitude. So what is the correct interpretation of wavefunction?
I am considering $\lvert \psi \rangle$ (arbitrary state vector) as being different from $\psi (x)$ (wavefunction).