If X and Y don't commute then their flows don't form a parallelogram like that. You need a pentagon that adds an [X,Y] side at the far end. Then both pictures are about how vectors rotate around loops, and the first picture comes about when you try to specify the loop by the flows of two vector fields.

The apparent necessity of left/right actions is sneaking in by assuming our only choice is an order, which is 1D automatically. Arguably the 'linear' parts of math stem from time, since function composition represents joining operations temporally (e.g. group actions). In fields that get away from "ordered processing" higher-dimensional structures/languages appear more often. I'm surprised no one has mentioned higher-dimensional algebra (all but named in the question) or diagrammatic algebras, where the points about up/down actions and the constraints of linear notation are well appreciated.

I don't understand why this is considered off-topic. If someone asked the exact same question but without the quote from a reporter it would be a legitimate question, would it not? It's about professional mathematics, at least one source has been mentioned that does name inventions of Grothendieck, and it's clearly answerable, even if mostly in the negative. The question is not about what the reporter thought, the question is about whether they were right, which is a mathematical question with a mathematical answer.

It's not against the rules. The questions on the site do need to be at a high enough level, and sometimes there's disagreement on whether or not a given question is appropriate. Don't take it personally, this happens on a lot of questions.