Itzjbean nailed it...It's totally irresponsible for a breeder to sell a baby bird that is not fully-weaned to someone so that they can hand-feed them when they are not professionals. Most of the time, in-fact, probably 90% of the time, when people buy a baby bird that is not fully-weaned and attempt to hand-feed the baby themselves, the baby either ends-up not weaning properly and develops severe neurological issues due to not weaning properly and at their own pace, or they die from either aspiration of formula into their lungs, an infection in their GI Tract due to improper formula or ambient temperature, crop-stasis, etc. So please, do not EVER attempt to buy a baby bird that is not fully weaned...
To answer your questions, first of all, "weaned" simply means that the bird (or dog, cat, etc.) is no longer eating any food from it's parents or whomever is feeding it, and is totally eating solid food on it's own. So ALL birds are "weaned", that has nothing to do with whether they are hand-raised or parent-raised.
Hand-raised baby birds are pulled from their parents between 2-3 weeks old, housed inside of a Brooder until their feathers grow in, and are fed hand-feeding formula on a strict schedule by their breeder. They no longer have contact with their parent-birds after being pulled. When you buy a baby that was hand-raised by it's breeder it's typically very tame, and will allow you to handle it right away, will step-up for you, etc. So if you're going to buy a baby bird, then yes, you want to buy a baby bird that was hand-raised by it's breeder, and that has fully-weaned and no longer needs to be fed formula at all.
A parent-raised bird is totally fed and housed/kept warm by it's parent birds, and is not fed at all by the breeder. Now this doesn't necessarily mean that the bird will not be just as tame as the hand-raised baby will be. What responsible, really good bird breeders who parent-raise their babies do is each day, from the time the babies are between 2-3 weeks old, they pull each baby out of the next-box when both parents exit the box to go and eat, drink, etc., they block-off the nest-box so that the parents can't get back into it, and they hold/handle/pet/cuddle each baby for a good 15 minutes or more at least once every single day. They touch them all over, talk to them softly, etc., just like they would if they were hand-raising them, they really are hand-raising them, they just aren't feeding them...****And this results in the babies being just as tame as babies who are actually fed by the breeder from a syringe...AS ITZJBEAN STATED, actually "feeding" a baby bird has no bearing whatsoever on the bird's tameness at all. Not one bit. What determines the tameness of a baby bird is the interaction with humans/touching by humans that the baby gets during it's first weeks of life up until the point that they fully-fledge and wean.
And yes, baby birds start to fledge (fly) before they fully-wean, so by the time the baby is fully weaned and eating on it's own, it has fully-fledged and knows how to fly...This is important, because breeders should never, ever clip a baby's wings before they fully-fledge, as this can result in the bird never being able to fly or fly properly.
So the bottom line is that unless you have a ton of experience, education, and ideally you've been mentored by an experienced bird-breeder on how to properly hand-feed and properly and gradually "abundance-wean" baby birds, you absolutely should never, ever, ever bring home a baby bird that is not fully-weaned and eating only pellets, seeds, veggies, etc. completely on it's own. It's too risky, there are a million things that can and do go wrong and if you don't know exactly what to do on the spot the babies will die quickly. And not only that, but until the babies are 5 weeks old you cannot let them go all night without feeding them every 2-3 hours, so that means you can't sleep for more than 2-3 hours at a clip...And for what? Nothing. Feeding a baby bird by hand will not "create a stronger bond' between you and the bird, as stated it can actually do the exact opposite.
Think of it this way: When a baby bird is hand-fed by it's breeder, and then after it fully weans it goes to it's new home with it's new owners, the baby bird bonds closely with it's new owner/owners, it creates a bond with that person or people who are caring for it, spending time with it, etc. It's totally forgotten about it's breeder who hand-fed it, right? Yeah, it's the same concept as a baby bird leaving the nest after it weans. Birds don't live with their parents for life, they are raised by mom and then they leave and live their own lives, just like puppies or kittens. So you're actually much better off (and safer) by buying only a baby bird who was hand-raised by a breeder, who is already fully-weaned, eating totally on it's own, and who you can just start bonding with, training, and enjoying as soon as you bring it home.