Yeah, that's true too. The thing is with squats is you really need someone who knows what they're doing to talk you through them correctly. Correct form is actually more painful and taxing than most 'cheats'; the problem is by cheating you're putting parts of your body under load with the wrong biomechanical position, and that's when people blow out their knees or backs.
I don't even know if you need to go with a trainer. The power lifting gym I go/went to (it closed down, but not before they filmed all the gym scenes for the new Rocky movie there) was filled with dudes who can squat 1000+, so they were more than willing to show me proper technique/the basics for free.
Good distinction. If you want to be strong for track, your workout sounds more appropriate. The way to train for peak single-rep strength requires alternating high/low rep scenarios, say a 3x15-rep week, 3x8-rep week, and a 3x4 rep week, with the occasional 'warmup - 1x12 - 1x8 - 1x1-2 to get a new personal best.
When I raced track, our coach would have us do two sets of twenty squats, with a weight that was challenging for ten reps. I went from 175 pounds to 190 pounds in six months, and was squating 265 for twenty reps.
google super squats, deep breathing squats, or high rep squats for more info.