The first link from above still refers to the hump theory where the air has to travel faster over the top than the bottom. It travels faster simply because it's being drawn towards a low pressure area created by the wing.
Lift is generated by accelerating air downwards, via the force = mass times acceleration. Coinciding with this downwards acceleration of air will be pressure diffentials on the wing. A bird can flap it's wings to generate downforce, like a hummingbird, or it can glide like a pelican.
The article also mentions that birds don't fly higher than needed, but doesn't mention the fact that birds will use thermals to save energy. I fly radio control gliders and quite often share thermals with birds. Predator birds will only thermal up to the point where they can barely see their prey. Here in southern California, the local seagulls will thermal the highest, going into the clouds, since their intent is to get as much out of a thermal as possible before gliding onwards to another location.
At some slope sites, where there is an updraft of air, you will find some birds diving downwards, and gliding back upwards in the updraft, apparently just for fun.
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