Do you think there's any difference in "I influenced the project's outcome" and "I controlled the project's outcome"? Because his statement -and meaning- seems incredibly clear to me. And if you don't think there's a difference, maybe that's the root cause of why I agree with him, and you disagree. As always, play what you like
We seem to have some "conceptual dissonance" when it comes to the use of language as we're again on opposite ends of things

Only this time, instead of fiat its the influenced vs control specrum.
I agree with pemerton here in the conceptualization of influenced as a spectra on the continuum of fully autonomous versus fully controlled. As your "influenced spectra" moves further toward the "fully controlled" end of the continuum, the weight of the influence (even though it may not be overt or physical imposition) becomes such that it drowns out the conscious aspects of personal autonomy that dictate behavior. The "imposition of will" is an enormous factor in this. There are a myriad of examples of this in real life but I'm going to stick to a few martial examples to try to illuminate the issue (and a real life anecdote that I don't know why I never considered before).
Consider these three dominant athletes of the last 25 years; Micheal Jordan, Tiger Woods, Roger Federer. What do they all have in common? Almost universally, their competition would wilt even before they got on the court/course and would specifically and inevitably wilt in the big moments. Yes, those three would raise their level of play but the wiliting of their opponents (and the surity of it ocurring before it even came into being) was a product of the imposition of will which, in their case, moved "influenced" much further along the continuum toward "fully controlled" without any physical imposition. The uniformity of opponents' wiliting shows that. Opposition and expert spectators would regularly comment on how the battle was won before it even began. That is "the imposition of will" that passively moves "influenced" along that continuum such that it manifests as a spectra infinitely closer to "fully controlled" than "fully autonomous" without their being any physical imposition.
My own personal anecdote which, amusingly, maps extraordinarily to CaGI (and I still don't know why I haven't considered it to date). I grew up in a "hard knocks" culture. If you were challenged, and you backed down, you were labelled with the P-word. Being labelled with the P-word has enormous consequences for that kind of culture. As such, you find yourself in physical conflict after physical conflict and it just becomes an institution. From the age of 7 through 20, I was in an enormous number of fights. It just came with the territory. If someone challenged/tested you, it was a borderline involuntary thing to accept it; and that was that. The last real scrape I can recall getting in was at a basketball court that I wasn't particularly fond of going to (the fight:ball ratio was horribly skewed toward the latter). I've always worked considerably hard at being a good sportsman and being honest. I'm the guy who will reflexively call a fall on myself if it occurs (for those who aren't aware of the dynamics, that is very rare). I don't like to win by way of cheating. So when I see people who do that, it infuriates me (and it really did back in the day). Three guys who were trouble (I had experience with them in the past) showed up. These guys are the worst kind. Unsurprisingly, I found myself covering the worst and most mouthy of the bunch. The game got more and more physical and he and I were going at it pretty good. All the while, I knew that if something happened, it wasn't going to be just he and I; his other two friends would absolutely jump me. I pride myself on being heady and pragmatic. However, thiings got really out of hand afterwhile and he got more and more mouthy and more and more gratuitously physical and literally trying to injure me. One one play I gave him what he had been giving me; I fouled him hard. I didn't mean to foul him that hard and I immediately apologized. Well, he wasn't in the mood for apologies, he was just looking for a fight (as he and the other two always were). Mid-apology, he takes the ball and two-hand chest passes it right into my face. At this moment, I still knew, still had it in my mind that the moment I engage here, its going to be 1 on 3. My blood boiled. He backed up and started goading me and calling me alll manner of names (but to be honest, I barely heard hiim at that point). Furious at all of it and knowing that I couldn't back down (fully aware that in short order I would have the other 2 on me), I sprinted at him, closing the 20-30 foot distance and tackled him. Almost immediately, to no great surprise, I took a shoed kick the back of my head from one of his friends and the three of us were going at it. So there I was, a heady, pragmatic guy (and at that point much more than I used to be as a kid), knowing full well that what I was doing was "tactically foolish", but I failed my Will save and did it nonetheless.
I guess because of all of those things above, things like CaGI not only don't bother me, they make sense; not only in real life but certainly in the heroic action/adventure genre.