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).
- 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.
DMG Page 239:
Often, players ask whether they can apply a skill proficiency to an ability check. If a player can provide a good justification for why a character's training and aptitude in a skill should apply to the check, go ahead and allow it, rewarding the player's creative thinking.
Since the players have justified using both Arcana and Religion, then let them use either. Both skills are perfectly good ways to have acquired the knowledge.
Best Answer
This is not explicitly covered
By RAW, on a successful Wisdom or Intelligence check, there's nothing that indicates the Ranger gets more info than any other class. The Favored Enemy feature description states:
What you get is easier access to the info by virtue of gaining advantage on intelligence checks.
However the description of Favored Enemy also states:
While this is flavor text, it does suggest that the ranger has considerable knowledge, moreso than other players, of their favored enemy.
I've approached this in several different ways as a GM:
Ask the player
Sometimes I simply ask the player what information they want to recall and then they roll to recall it. This can be any info they choose about the enemy from its stats or description. If they fail the roll they can't try to recall that same info until the next day. However they can try to recall different info on the same day.
GM's choice
Sometimes I just choose as GM which info to give. When I do this I give 3 pieces of basic info, like creature name, AC, and HP (+/-20%) and then an additional number of pieces of information equal to Int mod + proficiency bonus. For example a 1st-level ranger with 12 int would get 1 + 2 additional bits of info of my choosing. I try to keep at least some of the info useful in the moment, but might also include quirky bits of info.
Consistency and fairness
It feels to me most fair to the players if the GM chooses a method and sticks with it for the entire campaign. Also, I like to have some part of the mechanic encourage me as GM to give more info than I normally would, otherwise there doesn't seem to be any benefit to the favored enemy feature.