Fake password reset links are also a common attack vector, so yes people are told to be also cautious of those.

Otherwise it's been a while I haven't seen an reset link instead of a reset code. Copy/pasting is not much of a hassle, and it works even if the mail is checked on a different device.

The only real link I had to deal with were app callbacks that were explicitly labeled as such (with instructions from the app to explain what to expect)