First up, you'll never see "leviĝi min", that's wrong. Pretty much any time you thinking "-iĝ- ... -n" it'll be wrong. Once you've got -iĝ- in there, you won't have a direct object following it.
Now for your example: • "Mi levas min" does literally mean "I get myself up". • "Mi leviĝas" means something like "I am got up" or "I am raised up" but without actually mentioning who or what is doing the work of getting you up. It may well be that you're getting yourself up, but the sentence could equally apply if someone else is helping you up or even if you're passively allowing someone to bring you to a standing position.
And so it is with "La arboj kliniĝas". All that tells us is that the trees are being bent over, but with no information about whether that's because of the wind, some snow, or because the trees have all died off and are wilting like dead flowers under their own weight.
Adding "...pro la neĝo" to the sentence tells us that it's "...because of the snow."