Crouching down to its height could expose you to worse injury as you are now putting your face and throat much closer to the animal.
These are the instructions I give my children, who love animals and are used to dogs and want to pet every dog they see.
> It seems obvious that the more respect you show an animal in this situation, the better.
It depends. Some animals behave better when it's clear to them that a) they're not the alpha and b) the alpha has everything under control.
The core point is: move slowly. Hectic movements will induce stress in any animal. Give it time to react to your presence and try to look at the body language (especially if it tries to back away from you or growls/hisses at you)... how often people fail to respect a hiss, it's unbelievable.
It's amazing how many people don't know that rule #1 for a cat is: let the cat be a cat, if she wants to interact with you she will come to you and ask.
Even more amazing is many people don't know that this is also valid for dogs - no, you should not complain that a dog in a subway nearly bit your hand off, when you simply go to the dog and pet it without asking the owner if the dog actually likes being touched by random people and don't first hold your hand in front of the dog's nose so he can sniff it before touching the back of his head from behind. That often enough the dog pays with his life for your inability to respect boundaries is horrible.
Then again, people generally don't respect the personal boundaries of anyone they deem "incapable" of having boundaries, including kids ("give stinky aunt a kiss"), women or old people... so it doesn't surprise me...