Take strength in D&D5ed for instance. It is claimed to represent "measuring physical power". What it does is providing a modifier on checks the humans involved associate with the representation, affecting the probability of succeeding at a tasks.
If I were to guess at the functions of this mechanics I would suggest at least these 3:
1) A persistent characteristic of a character, informing the success rate of certain tasks over time (Informs the "simulation")
2) A player motivator to have the player in with high strength character engage in more strength related tasks/challenges. (Individual behavior priming)
3) A in-group role differentiator, giving the player with high strength character increased spotlight in situations where strength is beneficial. (Group behavior priming)
One thing to note is that these match up quite well with the fiction. We would expect a strong person to succeed more in strength related tasks, be more willing to engage in them, and step up to the task if the group need strength.
Now imagine a fictional game that is exactly like D&D5ed except the following: When declaring an action, both task and intent must be specified. If you roll equal or over dc, task succeed and intent happen. If you roll less than DC the GM has to narrate something really dramatic that brings the story forward. GM are in this case free to decide if the task succeed or not, but the intent at least isn't fully acheived.
If we now look at the mechanic of strength, we still have the same claim regarding what it represents. But what it does has changed. It no longer (purely) affect task success probability. Indeed it more strongly correlates with intent achievement. It also strongly affects the probability of dramatic stuff happening in situations where strength is involved.
How does this affect the functions? Depending on the motivations of the group 2 and 3 might still be relevant, but for groups that really seek mayhem and drama it could have the oposite effect! And the first function is almost completely obliterated, as a weak character that inspires the GM to think of good succeed with complications might very well succeed more often in strength tasks than a high strength character that inspires the GM to think of more failure with twist scenarios.
This is where my claim about FF/intent on success games come from. The most likely function I can see for strength in this scenario (also informed by how people talk about how these games play) is as:
1) A narrative marker indicating what the player want the perceived value of the characters attempts to use strength should have in the fiction (Informs the narrative)
I think this still is still reasonably associated with the given in-fiction representation, especially when genre conventions is taken into account. However a trad player is used to the representation of physical strength to be used for a different set of game purposes.
This is why I think it can be instructive at least in a transition period to not think of the skill/strength in terms of inate ability (due to the associations with "wrong" game functions), and rather think of them in terms of the function itself.
I think this analysis while not perfectly matching any existing games, nonetheless is instructive in where actual differences lie.