It clearly depends on the ripeness of the banana.
On the initial opening, if the banana is a bright yellow, or slightly green (I actually can't tell the difference), then you need to bite the top to be able to make a clean peel without mushing the top.
If the banana is starting to ripen nicely with lots of blackening of the skin (not bruising, but ripening spots), then there's usually a a nice black split waiting across the top of the skin anyway which makes it easy peel.
From then on, the eating is also different.
A riper banana is more likely to break in the middle, so eating without the skin is a trickier proposition. Getting the full peel without broken banana disaster requires a lot of skill. However, if you can manage it, it's the best way. Holding a riper banana in the skin while you eat and peel will result in a much mushier banana by the time you reach the bottom.
For a firm banana, the peel and eat as you go along technique works the best. However, you're missing out on the perfect opportunity of sliding out the banana from the skin with minimal damage. Also, a stiff fruit is much less messy in the hands.
Personally, I like to always do a full peel, pinching the bottom of the banana to remove the tough clitoris (it always tastes a bit nasty). If possible, I like to bite the top anyway. The bitterness sets up your taste buds and makes the banana taste much sweeter.
I find it odd that cartoon monkeys are always portrayed as peeling a banana neatly. I've seen monkeys and apes eat them, and (excluding humans) they never, ever peel them!
Bananas should never, ever, EVER be cooked! Ever (you're already wrong Neal, don't make it worse for yourself
). The best way to serve banana is sliced longways with ice-cream in the middle as a banana split/longboat (depending on how Danish you feel). Secondary is in bread, sliced as a sandwich. Tertiary is mixed with porridge. Nom!