Rules for bribes when "Asking For Help" in turn vs bribing outside of a turn
There's nothing in the second rules quote that contradicts the first rules quote. Regardless of in-turn versus out-of-turn, giving an item to a person requires that the item is in your inventory (it's on the table in front of you, regardless of whether it's equipped or not). 

This doesn't mean that's the only way to affect a combat or bribe your opponents, of course: you could threaten/promise to play a potion, one-shot item, or monster-enhancer on the current combat (or someone's future combat), either for or against the player(s). You wouldn't be giving the player an item, but you would be using the item in a way that could affect their current decision. There's nothing requiring you to follow through on a promise, of course, but it's going to affect your reputation: if you don't follow through on a promise, your fellow players probably won't believe you the next time you promise something.

Some clarifications that may help:


Equipped items are face-up in front of you (i.e., part of your inventory), and they add to your Levels because you have sufficient slots to use them. (Potions and some other items are slotless, but most other items take a slot: one hand, headgear, armor, etc. Potions can be played from your hand or your inventory.)
Carried items are in your inventory; they're whatever you're not equipping. You don't have to drop them, and you don't have to apply Charity to them, but they're not adding to your Level. If you have three one-handed items, then you're going to equip two of them and carry the other one. I usually turn carried cards sideways to make it clear what I'm equipped with.
Your hand is not your inventory, and is unrelated to your equipped and carried items. It's always referred to as "your hand" in the rules.