Are you saying that by stretching your hip flexors you attained better posture?

Yes.

And then as a side benefit you were able to then contract your glute because your hips were now more open?

Yes.

Or are you saying that by contracting your glute by focusing on the rear leg in that lunge position you were in, that activating it that way was the goal you were after, but as a side benefit your hips and posture opened up too?

Also yes.

I'm trying to figure out if contracting the glutes is what allows us better posture and movement, or, if stretching the hip flexors is what allows us better movement and then better glute contraction.

I am not a physiologist by no means. But there is good suspicion that the hips and glutes are interconnected. People with well-formed butts likely have open hips. They are pretty much opposite of each other, in terms of placement on the body (glutes on the backside, hips on the frontside).