Virtual Ground Paradox?
It's #2.  For a "perfect" theoretical opamp, the open-loop gain is infinite, and this makes the difference at the inputs zero. When introducing opamp circuits, or when working out how things are supposed to work, people normally think about the "perfect" opamp.

When thinking about the performance of a circuit, we usually have to start thinking about the imperfections of a real opamp. For a real opamp, the open-loop gain is not infinite, and there is some difference between the inputs. To take the example of an LM324, the open loop gain is about 115dB. That's a little less than a million volts/volt, so if there is a 1V DC output, then the inputs are different by about 1uV.  Most of the time you can ignore that.

It gets more complicated for AC. At higher frequencies, the gain drops. For the LM324, it goes to 0dB, i.e. 1V/V at about 1MHz.  At that point, the inputs certainly will have a large difference.  Practically speaking, the amplifier just doesn't work any more.  For frequencies in-between, the gain of the amplifier (inc. feedback) will vary. The term "Gain Bandwidth Product" is used to describe what gain you can have at what frequency for a given opamp.

This is just one of many imperfections a real opamp has.  Another very relevant one is input offset voltage. This is the difference in inputs which results in a zero output, and it's not always exactly 0.  This might be more important than the limited gain in many cases.  Other imperfections you might want to consider are saturation/clipping, input current, PSRR, CMRR, nonzero output impedance and many more.