pathfinder 2e – Is a natural 20 always a critical hit?

No, a natural 20 is not always a critical hit.

The key phrase is in the second passage you’ve quoted.

If you rolled a 20 on the die (a “natural 20”), your result is one degree of success better than it would be by numbers alone.

This turns a critical failure into a failure, a failure into a success, and a success into a critical success (leaving a critical success as a critical success).

The first passage says “when you succeed on a natural 20,” which is to say, when you roll a natural 20 and the result would be a normal success before factoring in the increase in degree of success.

So, a natural 20 can even be a failure if it would usually be a critical failure, it’s just that this is very uncommon in the game due to the expected level of enemies you’ll be facing.

EDIT: To answer your final question specifically,

So if I roll a 20 on my third attack, but it is still lower than the enemy AC (because of MAP for example), is it a critical hit, or a normal hit?

It would be a normal hit, unless your roll total would otherwise be a critical failure, in which case, it is a normal failure.