[RPG] What type of damage accounts for bleeding damage

The primary source on Favored Enemy is either Player’s Handbook in the ranger class description, or Rules Compendium if you buy its assertion of primacy. The descriptions in the Invisibility description in Dungeon Master’s Guide, Improved Manyshot in Epic Level Handbook, or even darkness, despite also being in the Player’s Handbook, are definitely not the primary source on Favored Enemy.

Further caveats, limitations, and addenda not mentioned in the primary source description are contradictions with that description. If Favored Enemy had defined itself as precision damage, the primary source on precision damage would apply, but it didn’t, which means that neither the precision damage description nor anywhere else can define it as such.

I would be inclined, in general, to follow contradictory rules as far as they go. Darkness and Invisibility cannot define Favored Enemy as precision damage in general, but it can say Favored Enemy doesn’t work in those conditions. Here, specific-trumps-general: rather than trying to redefine what Favored Enemy is (in which primacy asserts itself), they can define a special case which acceptably contradicts the general rules. So my reading of the rules as written would be that Favored Enemy is not precision damage, but it does fail to work in cases of darkness or Invisibility, and does apply only once to Improved Manyshot.

Ultimately, however, I would mostly ignore any and all rules as written that apply any more needless limitations on Favored Enemy. Of all the iconic core class features,¹ Favored Enemy is one of the weakest, and that’s even assuming that it “just works” on any and all attacks against the designated foes (including, therefore, Improved Manyshot).

1. ^{Slow fall and wild empathy, if counted as “iconic class features,” are definitely weaker than favored enemy, and smite evil and trapfinding give it a run for its money too. That’s still a small list compared to all of the options.}

PHB p.197

Resistance and then vulnerability are applied after all other modifiers to damage. For example, a creature has resistance to bludgeoning damage and is hit by an attack that deals 25 bludgeoning damage. The creature is also within a magical aura that reduces all damage by 5. The 25 damage is first reduced by 5 and then halved, so the creature takes 10 damage.

You take off the damage reduction and then halve the damage. Option 1 in your question.