Is there a standard/foolproof way to tell what input or output impedance is?
pjc50 summed up the solution in comments.

Input impedance is defined as the impedance you would see no matter what the input voltage/current. With the fact that ideal op-amp analysis pins both inputs of the op-amp to the same voltage, and one of the inputs pinned to ground, the other terminal can be treated as a virtual ground. Now you have an infinite current source/sink at that virtual ground. That means if you create a test input voltage on the input side of the circuit, all it will see is 10k to "ground". 1V/10k=100uAmps. Input impedance is the test voltage divided by the resulting current so 1V/100uAmps = 10k. So that's how they're coming up with 10k as an input impedance.

If you wanted to determine the output impedance, you could look at the tail end and try and inject a 1V supply at the output side of the op-amp. The output of an op-amp can source or sink infinite current (pseudo-ground) so you basically have a connection from the output wire directly to ground. A wire directly to ground is 0 ohms so the output impedance in this case is 0 (or very low).

From what I can tell, your confusion comes then from not understanding how an ideal op-amp works. The input side of an op-amp tries to pin its two inputs to the same voltage and the output can source/sink infinite current to create the desired voltage.