Player skill vs character skill?

Under BX/BECMI, it's not a d20+ mod, it's 1d20≤ Att for many checks.
In Moldvay B/X Basic the only example I can find is an optional rule on page B60. Everything else is 2d6 based (reaction rolls, morale) d6 based (listening, finding traps or secret doors), or percentile (thief skills) or more frequently the DM just decides. The one optional example is right after a longer section about a DM just deciding and then optionally assigning percentiles for success.

"That's not in the rules!" The players will often surprise the DM by doing the unexpected. Don't panic. When this happens, the DM
should just make sure that everything is done in the order given by the outline or sequence of events being used. Minor details may be made up as needed to keep the game moving. All DMs learn how to handle both new ideas and unusual actions quickly and with
imagination.
Quite often a DM can decide on a solution to a player's actions not covered by these rules. Other times, a problem may have no simple solution. One quick way for a DM to decide whether a solution will work is by imagining the situation, and then choosing percentage chances for different possibilities. For example, suppose the DM is running a combat that is taking place on a ledge next to an unexplored chasm. One player suddenly decides that his character has no chance to survive combat. The player announces "My character wants to jump into the chasm to escape!" There may be a chance that he will fall to a nearby ledge or land in a pool of water at the bottom of the chasm. The DM thinks about the dungeon for a minute, and remembers that an underground river flows through some of the lower dungeon levels, so there might be a pool below. Even so, the character will fall 60', and a normal fall will do 1d6 points of damage per 10' fallen. This character has only 7 hp, and seems likely to die even if the water cushions his landing and reduces the damage. However, there should always be a chance to do something nearly impossible. A player should have, at the very least, a saving throw or a stated percentage chance of a miraculous occurrence saving the character. The DM answers: "Looking down into the chasm, your character can estimate that he has a 98% chance of dying, no saving throw, if he jumps. If you decide your character jumps, roll percentage dice. A result of 99 or 00 will mean that your character lives, but any other result will mean that he will die in the attempt. Do you still want to jump?"

"There's always a chance." The DM may want to base a character's chance of doing something on his or her ability scores (Strength, Dexterity, and so forth). To perform a difficult task (such as climbing up a rope or thinking of a forgotten clue), the player should roll the ability score or less on 1d20.

Because the Rules strongly imply I can. In BX, it's in the example of play that actions can be checked by dictating a roll against attributes.
I can't find any such example in either of the examples of play on B28 or B59. The PCs just say what they are doing and the DM adjudicates. They accidentally leave behind their dead companions gear and supplies when the players say they leave, no int or wis check involved. When trying to talk down the encountered hobgoblins they just act out what they say and there is a reaction roll (which would possibly be modified by Charisma and is modified by the chosen technique) not a charisma roll.

Are you thinking about the B60 section on acting on information the characters would not know? That is a bit different than int score rolls limiting planning.

"Your character doesn't know that." A player should not allow his or her character to act on information that character has no way of knowing (for example, attacking an NPC because the NPC killed a previous character run by the player, even though the NPC and current character have never met). If the players get careless about this the DM should remind them. The DM may, in addition, forbid certain actions to the characters involved. The DM should make it clear to the players before the adventure begins that characters may not act on information they don't have. It will save lots of time later.
 
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So back to the question. Where are your lines on player skill vs character skill?
In general, apply each where the fun lies.

In the case of traps, I'd suggest building in things like tells and other clues for the players... but if you find you have a group that just hates that stuff, default to rolling the dice. (But I'd suggest going with one or the other - if the group does enjoy traps as a challenge to player skill, adding the dice as a backup is likely to make it all too easy.)

Likewise, most players who create outgoing, social characters will want to handle the interactions themselves - default to player skill. But if you have a socially awkward player who nonetheless wants to play such a character, that's fine too, and in that case let the dice handle the details. (And, again, one or the other; not both.)

And so it goes.

Unfortunately, that all seems to be a real weakness of D&D 5e in particular - very often a player who is interested in a particular aspect of the game proceeds to create a character who is good at that aspect of the game... and is 'rewarded' by the mechanics utterly trivializing that aspect of gameplay. (Or, at least, that's the case in the 2014 version - 2024 may do better in that regard.)
 

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