In this context, where the trainer is distinguishing between flexion, bend, and being on the aids, yes, this is the correct use of the terms.
Your professional is describing lateral flexion of the jaw (moving the head sideways without the body changing shape), which for some trainers is done as a discrete exercise, in hand and in the saddle, standing or moving.
However, yes, “flexion” does have a larger dictionary meaning that something is bent. So it would also be correct to use the term “vertical flexion” to refer to the horse giving at the poll, that is bringing the nose closer to the vertical. And of course you could refer to anything that’s bent as “flexion,” as when the vet does a PPE exam by bending the knee tight, releasing, and seeing if the horse is sore after: that is a flexion test.
It is also true that the expression “on the bit” is a mistranslation of the original phrase that meant “on the aids.”
On the aids means that the horse is collected, attentive to everything that is being asked, and has vertical flexion at the poll. Horse doesn’t need to be vertical with the nose, and certainly shouldn’t be behind the vertical, but does need to have “given” his poll to the rider. The phrase “on the bit” makes people think that the goal is primarily tight rein contact, when in fact a horse can be on the aids beautifully even with a slightly draped rein.