great question, and I certainly understand where your groups confusion came from! There are a couple of paths I could see your group taking depending on the situation and the players' intent.
First of all, your group's solution to the problem is totally valid! It's one that I would certainly have presented to the group. As players, your are the most important resources for finding a way forward in these edge cases - especially in the moment of play. Expert was definitely triggered, and I applaud your Intellect players instinct to enhance the Infiltrator's action, not supersede it.
The confusion comes from how some Moves are named, the criteria for more than one trigger being met simultaneously. Expert of course, as you have mentioned, and Share Expertise - but the Intellect player's intent also points to Lean on Me being triggered.
I usually approach these sorts of situations by looking at the player's intended outcome, fictionally and mechanically. Your group did this by having the discussion, and saying they wanted to give the Infiltrator Advantage. Expert clearly does that - if you can trigger a Move. Share Expertise interacting with Expert can do this. The Intellect shares what they know about programming or this particular program as the resolution to the Share Expertise roll, then as Expert interacts with it, they choose to give the Infiltrator Advantage.
The other way I could see this playing out is with Lean On Me. If the Intellect player really wants to play on the angle of helping the Intellect & their relationship as the point of tension, the Intellect could roll Lean On Me, with the intent to use it to help the Infiltrator - probably by giving them Advantage. Expert would influence this roll by allowing the Intellect to choose 1 more or less option from the list in Lean On Me.