I think a classic automatic transmission needs hydraulic pressure to engage the bands. And since the gear pump is on the input shaft side it doesn't run when the car is pushed or towed.

In this case, though, an automatic transmission is - for all intents and purposes - a transmission that does not require the driver to handle shifting at all. The DCT fits in this category.

AT is "the car changes gears".

Sometimes a distinction is made between AMT and others because AMT is kinda shit, but both DCT and torque are AT.

Also modern torque converters have lock-up clutches, meaning once they've switched gear they've got the same direct-coupling efficiency as a DCT. The ZF8 mentioned by Klonoar is a torque converter.

Ha, you're right! I don't have the energy to go edit my parent comment but hopefully it's clear (or at least, now) that I meant old school vs modern AT transmissions.

Thanks for the correction!