I'm not a native English speaker but I think in this case "in the tree" is the way to go: I imagine you could see a tree as a collection of branches, leaves etc and if the outer leaves mark the "boundary" of the tree, chances are the bird will be somewhere inside that "box", thus IN the tree (unless it sits on the very top branch of the tree I guess). It's the same in Dutch by the way as we say "de vogel zit in de boom" (although that may not help you hehe ;))
If you're sitting on a tree (e.g. you're a lumberjack and are having a break) I think you'd say "han sitter på et tre".
US English speaker here, you are correct. If you said you were sitting on a tree, I would assume that the tree has fallen down and you were sitting on its trunk.
This is how I use in/on for trees in English:
If something is part of the tree, such as leaves, fruit, or acorns, then it is on the tree.
If something is not part of the tree and is within the branches, such as an animal, person or a tree house, then it is in the tree.
However, if something is on the trunk, not in the branches, then it would be on the tree.
For example, a birdhouse can be on the tree (trunk) or in the tree (within or hanging from the branches).
In English "up in a tree" is just an alternative way of saying "in a tree", sitting on a branch of a tree. It emphasizes that it's above the speaker (for example, you wouldn't say it if you were in an airplane looking down!), but it does not necessarily mean that it's in the highest part of the tree.
I think, that "er" mean "is". But, we do not use it with the verb, like: "Jeg spiser et eple." We use it with the location: "Et eple er på bordet." and with the easy sentences like: "Det er ikke et eple." I hope I help you and you are not more confused. :-) Sorry for mistakes, but I am not native speaker. :-)