Good question, I thought it was. Never had a problem until recently. I kept my elbows mostly straight (as opposed to flared out) and brought the bar down to around the sternum. Hands around shoulder width. The last couple of years I've slowed down so nothing is herky jerky, to focus on the muscle. Was doing great and then over the past six months, this flared up.
As for pushups, keeping palms about shoulder width apart and at chest area. I do feel something in my tris (good that is) but also in the chest, i.e. results. What I wonder is since pushups use less "weight," should I do more, do them every day...or rest like with the bench. Usually I do a few sets, stopping when the muscle is tired, but nothing crazy - I am used to heavy weight and fewer reps. (Not an Arnold, just what I'm used to.)
I do advise a coach, any sort of free weight barbell movement is highly technical with many tiny details at every stage of the lift. I used to do heavy 5x5 sets, but my chest didn't respond to this & I knew I was doing it wrong. So one time I took 6 weeks to do 3 to 4 sets of 10-12 reps (around 60kg to start with, later around 70kg) every day, focusing on every rep. You learn the movement well, and my chest finally felt it because I started using it more, instead of shoulders, due to the better form and "focusing" on my chest more.
You can bench or squat "wrong" but it works for a while and even for decent weight, but once the weight gets heavy enough, the "wrong" muscles you end up depending on too much can't handle the load. eg: you front deltoid can't handle nearly as much weight as your chest, and your biceps can't handle nearly as much weight as your back; I plateaud on pull ups at around 10 reps, but later found out it was all biceps, working on form now to make it more lat/back, but at this point my lat/back is comparatively underdeveloped, so my max reps are less, but with time these muscle groups have more potential for more reps or adding weight.
I'm no gym rat. Lift mainly as prehab and to have some mass on me. Has prevented some injuries from sports that involve falling on the floor at high speed head first
but this means I tend to focus on form more than weight, as I have no incentive to lift more, just to lift right. Have never had an injury in the gym, best squat is 130kg (belted) at 80kg, best DL 140kg (no belt, no straps) at 80kg, best bench 95kg at 80kg. Which is nothing crazy on the internet, but irl there aren't many doing that (specifically the squat and DL) in an average gym, and especially no one who isn't either a weightlifter or bodybuilder. I'm a runner/mountain biker, cross country skier in winter. Could I lift more? Totally if I prepped for it and really went all gas no brakes, but it's not worth the potential injuries.