Not quite, irrelevant, in fact not at all. The english seat for trotting is a forward seat, not even close to the seat needed for western riding a gaited horse. You want to see a good gaited horse start trotting, just ride them in English style, and you can ruin their gait in a heart beat. This is one on the biggest transitions you have to make when switching from trotting horses to gaited horses. We raised and trained trotters for 40+ years, then switch to gaited. BIG transition, we ruined a bunch of them before we found out why, forward seat is not a "NO" with gaited but rather a "HECK NO". I like to call the gaited seat, a chair seat, with the legs much more forward of the seat than english. You ride on your seat pockets, so to speak.
That's why I pointed out that english lessons are at wrong end of the riding spectrum (one end, forward seat, trotters; the other end, chair seat, gaited).
Every wonder why so many folks that have ridden trotting horses for so many years, can not get a good gaited horse to gait. Most of it is in the seat.
There is no such thing as a "trotting seat," just as there is no such thing as a "gaited seat." There is just a "seat" suited to the discipline.
I commend to you Dr. Deb Bennett's book Conquerors: The Roots of New World Horsemanship
. She breaks down the two basic seats as a la jineta
and a la brida
The "brida" seat is what we call the "chair seat." It was the seat of the mounted knight, the bronc rider, the roper. It puts the feet forward and the butt on the cantle. It allowed the rider to absorb the shocks of fore and aft motion (like getting hit with a lance or taking the strain of roping a steer). It also allows a very "noble" presentation, as by hollowing the back the rider can get more front end action.
The "jineta" seat means the rider sits in the center of the horse, balanced over their feet. It permits a high degree of both lateral and forward movement. It permits quick movements without the rider getting unseated. It is the root of the Sally Swift's "centered riding system" (which she copied, almost verbatim, from the "Ft. Riley seat" as taught by the U.S. Cavalry School, which they copied almost verbatim from the French Cavalry School at Samur). The horse is collected, but not rollkured or overbent. Lightness and action are the watchwords.
A more modern variation on the "jineta" seat is the Caprilli, or "forward" seat developed by Capt. Caprilli of the Italian Cavalry. This was developed to permit better cross country performance as it permitted the horse to jump much more freely and achieve higher levels without unduly hazarding horse or rider. If you're going to jump your gaited horse you're going to do it in the forward seat.
You can ride a gaited horse jineta, brida, or forward. Ditto for the trotter. It all depends on your job.
If you ride a la brida
you are going to be traveling hollow. That makes for a smoother gait (in the laterally gaited horse) but the horse's back is paying a price for your comfort. Done in moderation it's not a problem. Done to excess it means a really sore horse.
Again, the seat you select is driven by the job you want to do.