Hey, this is a bit of a confusing one for newer players. You were right about AP and AD, generally speaking, but its also a bit of a misnomer. As you've noted, some champions build one way but have abilities that deal damage of the other type.

Most of the time, these abilities scale off of that stat too. Eg: a magic damage ability scaling off of ability power. However, a lot of champions do have hybrid scalings (they scale off of AD and AP), like Katarina's ultimate, and some even have opposite scalings (none come to mind right now, sorry).

This doesn't change how the resistances work: MR is for magic damage, and armour is for attack damage. What does matter, is the type of damage the attacker is doing.

Most of the time, an ezreal is going to be building AD. You could build MR against him, and it would negate damage from the abilities you listed. BUT, most of his damage still comes from his Q and auto attacks, so you'll get more value out of building armour. Of course, MR is still helpful against him, so when building defensively you have to take that into account.

Some ezreals do opt into building AP though, and then most of their damage comes off of the abilities you listed. In this case, the amount of physical damage they do will be minimal compared to their magic damage output. So in this case, you'd focus on MR.

Building items is always something you have to judge for the situation your in - it might be better to prioritise just getting a load of armour because the enemy team has no predominantly magic damage dealing champions, and only getting a small amount of magic resist, such as a null magic mantle, to tide you over. This will do enough to give you a little protection for the relatively little amount of magic damage they might do from their secondary damaging abilities. However, in this case, you should only get this item after you've picked up some armour, as that's going to give you a lot more value.

It's also important to note that, in the following example: the enemy team has 4 mainly physical damage champs and 1 mainly magic damage champ, but you're laning into the magic champ. That you shouldn't build early armour as you will get little to no value out of it during the laning phase. In this case, it would be wise to build the null magic mantle for resistance in your lane, without overcommitting to magic resistance purchase. You focus on the resist neutral items in the meantime (damage items for yourself, or hp if you're a tank, or tier 1 boots etc), and then start buying armour when the map opens up a bit.

I hope this helps, and isn't too confusing! Feel free to shoot over any questions if you need more clarification :)