Can someone explain to me the difference between Attack and the side values attack of each weapon? like I can have one sword that gives me more Attack while a lower level sword gives me more attack to the sword attack value on the side...which is more important?

The one on the side with the sword icon next to it is your sword affinity, which is part of your overall attack. The one that's labelled attack is more important.

1 Affinity = 1 Attack when that type of weapon is equipped,

In other words, total attack is better assuming all else is the same. (Generally, the one with better skills is better though, if they have similar attack)

I was under impression that "weapon affinity" serves like a damage multiplier of sorts, to give emphasis about the weapon specialization mechanic of the game.

What bothers me more is the existence of some affinities that aren't listed in the Elemental/Weapon affinity list (Pierce, Blunt, Slice, etc...) and how exactly they affect damage, aside from just being another "strong/weak against" traits for monsters.

You're somewhat correct. Attack is the multiplier.

Affinity is just a method by which you can increase attack with 1:1 scaling. At or before level 100, affinity will cap at 3000, and adding in any of the base stat used to increase that affinity will no longer give you any damage.

At 105, the affinity cap is raised to 3500.