I think the idea of "safe" has been pretty well discussed, but I would like to approach your situation from another angle as well.
You mentioned that you have felt comfortable with the lesson horses you have ridden. You recognize the fact that this was in the controlled environment of an arena. Also realize that these horses have probably been ridden by many different people and under the guidance and support of an instructor.
You seem to comprehend things better than a lot of people. Many people take lessons and think they have learned how to handle horses. Then, they buy a horse of their own and never give a thought to asking assistance with this horse.
In group lessons, horses often do what they see the other horses doing. This is much easier for them than trying to figure out what the person on their back is trying to tell them. In private lessons, the horse is often responding to the voice or body language of the instructor they understand rather than the rider who is talking in a different language or at least a different dialect than they are use to. I speak of language figuratively to describe variances in cues due to the size and shape of the rider as well as balance and other factors.
While an instructor will not know a new horse you may buy as well as the lesson horses he or she knows well, an experienced instructor can be very helpful in getting you accustomed to any new horse you may purchase. Think of this as a language coach helping you and your horse learn to communicate.
This is a fair point, and a really good one. I'm planning on keeping on getting lessons once I buy a horse, I've just been frankly shocked at how many people will declare a horse 'safe' even after it bucked them off and bolted (and not for any good reason, either, just because it didn't feel like being ridden). How on earth can you think your horse is safe if it just turfed you off and hightailed it?
I go with my gut when it comes to horses. I've refused to get on more than half the horses I've looked at, which doesn't always make you popular especially when the seller is insisting how wonderful their steed is. And the one I did get on against my better judgement was the one I fell off.
As for the lessons and language thereof, yes, I have ridden very well versed lesson horses, but only in private lessons and without many cues from the instructor. (My instructor likes to sit back and watch from the sidelines, so unless there's telepathy involved, the horses aren't getting too many cues aside from the odd 'hurry up'.) So I am hoping I know how to cue a horse properly by now, but I am also looking forward to learning to speak a common language with my own horse.