1. The Exercise Buddy Personal Trainer
This type of personal trainer typically does the exercises with you. You are basically paying them just to do the exercises with you, but you're also paying them to help motivate you to exercise more.
Some clients prefer this style of personal trainer because they like the exercise buddy experience. It is more informal and friendly.
However to save money you could literally just find a real exercise buddy who has a similar schedule to you. Or several exercise buddies. Or find a group of people who like exercising together (outdoors usually).
2. The Drill Instructor Personal Trainer
Some people like having someone boss them around and tell them what to do. This doesn't necessarily mean they are shouting at you and ordering you about like a real drill instructor, but the concept is basically someone who tells you what to do (although not necessarily how to do it, see #3 below on how to do it).
Not everyone is into being bossed around all of the time, but every personal trainer has to do this at least a little bit because it is part of the whole teacher-student role to sometimes be telling people what they need to be doing.
3. The Form Oriented Personal Trainer
This type of personal trainer is focused on making sure you do the exercises correctly and is watching you to make sure you do them properly. They are also sometimes demonstrating how to do the exercises properly so you get a better idea of how to do it.
The Form Oriented instructor is really focused on teaching you the proper "how to" of each exercise, often for safety reasons so you don't hurt yourself, but also for efficiency reasons so you are getting the most out of the exercise instead of developing a sloppy form that is both inefficient, but also unsafe.
There are also legal reasons why personal trainers should focus on form and safety.
I recall years ago hearing about a "celebrity personal trainer" in the USA who was more in the drill instructor variety of personal trainer with respect to weightlifting, but she pushed her client too far physically and he ended up complaining about pain in his arms. She told him to "man up" and "no pain no gain", etc. He ended up tearing the ligaments in his arms and ended up with permanent damage to his arms, and consequently suing the personal trainer for negligence. It was wrong for her to be pushing her client so hard like that and to be ignoring safety issues. Nor are such lawsuits uncommon. Just do a search for personal trainer negligence lawsuit and you will find that a lot of personal trainers are failing to do their due diligence when it comes to safety issues with respect to form.
What about Me?
As an archery instructor (and boxing instructor, swimming instructor, and ice skating instructor) all of the sports that I teach are form oriented. Archery is extremely form oriented, but the other sports place a lot of emphasis on form too.
If I had to choose therefore I would say I am closer to being a Form Oriented instructor than the other two. Yes, I am still demonstrating things to my students, but I am certainly not the exercise buddy who is doing the exercise next to them. I spend most of my time watching the student do the tasks set for them and then correcting their mistakes. Likewise I do have to boss my students around regularly, but I also sometimes give the student the option to choose what we are doing on a particular day.
Eg. Let's say a student signs up for 10 archery lessons. The first five lessons I have a lesson plan for, but the the final 5 lessons are more dependent upon what the student wants to learn and what the student is more interested in learning. Thus I take cues from the student and customize the lessons to their needs.
I am probably somewhere in the middle, closer to average, when compared to most personal trainers, but with a stronger emphasis on the form oriented issues.
I firmly believe in the whole "safety first" mantra with respect to exercising and sports.
Take swimming for example. People need to learn how to swim in the shallow end of the pool before you toss them into the deep end of the pool and expect them to learn how to tread water. If a swimming instructor forced a swimming student into the deep end of the pool, told them to tread water, and then the student drowns I would fully expect the personal trainer to be found guilty of negligence and manslaughter. Not just sued. Imprisoned.
The "safety first" mantra never hurt anyone. But a lot of people have no doubt been hurt or died historically because they ignored safety concerns.