Of your bird moves away then you are going too far. You don't want it to move away.
A bird bites for several reasons. Sometimes it bites to play, sometimes it bites before it's uncomfortable. If it bites for the second reason, it bites as a last resort. There are many subtle cues it will Gove off before biting. Some of them are leaning, raising its feathers or flattening them, pinning pupils, looking away, turning its head. If your bird bites, it's telling you that you aren't listening to it.
If your bird is biting for the first reason, then this formula won't work.
I DISAGREE WITH THAT STATEMENT... NOT ALWAYS!
For more than half a decade, I rehabbed "the biters." (I've worked with around 350 birds, not counting my own.) This I'm fairly well trained to speak on intelligently:
There is the normal fight or flight reaction which is fear biting;
There is birdie manipulation/practical jokes (My Ruby)
There is "I DON'T WANT TO!"
There is "I'm mad at you" biting. (You locked me up too long!)
There are tantrums. (Toddler emotional melt down. I AM REALLY UPSET RIGHT NOW!)
There are hormonal biting. (I'm horny and grumpy. Don't touch)
There are territorial intrusion issues. (Get out of here!!!)
There are hoarding/possessive issues. (That's MINE!!!)
There is displacement biting. (If you can't bite the one you want to bite, bite
the one you're with!)
There is jealous/overbonding biting. (He's MINE!/Get away from him/You're not my person!)
Worse, there is MATE AGGRESSION bird on human (overbonding a step further. Power and control. Do what I say, I'm your mate! Obey me! Don't make me hurt you!)
There is the bird "looking out for you, trying to protect you by warning you of danger" biting (You're too close to that scarey thing! Hey stupid wake up, it's gonna get you!)
There is accidental biting (i.e. playing too rough/All I was trying to do was hold on, but I don't know my own strength, so I hurt you... i.e. Woody almost taking my finger off holding onto my finger when I was scratching his head.)
There is "amazon overload" biting. (Altered states)
Then there are large toos and sennies that have lots of 'tude and MPD. Otherwise known as "BECAUSE I CAN" biting. (PSYCHO BIRD!)
I could go on, but this is already starting to sound like Bubba in the "shrimp" scene from Forest Gump. So, I will stop now.
My point is it isn't always a simple generalization, and they don't always warn you... in fact, ala my Ruby, sometimes they deliberately DON'T warn you. Cuz they secretly want to see you dance and shout. "That never gets old! It's so funny when they do that!"