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Thanks, your edit and explanation help a lot :)
To be honest with you, the look of the ear was kind of intended, it was originally meant to be a normal ear but when I saw it looked like a pair of leaves (or a butterfly wing as you said), I decided to leave it like that since I really liked it that way :P Also, the mouth was supposed to be open, but I guess I failed at making it look that way. But it doesn't matter, I guess it's better if it's closed.
I'll be sure to keep the stuff you told me in mind. I plan on making more sprites/pixel art with just 3 colors and with different color schemes, so I can practice my skills and improve like always.
Better!
The ear looks a bit like a butterfly wing but I can live with that.
Here's a sloppy edit + banding pointer: 
More pointy mouth, fixed trunk shading and folds, darkened back ear for readability. Tried to get those head lumps that I think are a main feature but failed, it would take a complete shading overhaul I guess. Anyway I left a lot of mess too, my pixeling is rusty ;)
Banding isn't of the worst obvious kind but there are still many cluster alignments, sometimes across large distances (I didn't point them all!). They can't (and shouldn't) be all removed, but looking for them and fixing what can be is a good way to refine clusters into optimal shapes.
Good to hear from a person like you :3 Thanks for the critique and tips.
I have tweaked the colors, head, trunk and shading on the belly. From looking at some elephant pictures, I figured out the head should be oval-ish, like an egg. And of course, I copied the visible ear before erasing the original head and trunk, since I think it was good enough and liked it myself a lot anyway. I also shaded the new head + trunk differently than I did with the original one, and personally I like it more. And, for some reason, I can't find where's the banding even though I know what it looks like, unless it's the outline or something.
Anyways, what do you think? Is there anything else that needs fixing/improving? :)
Great exercise for practicing pixel control, and you do good =)
The body is good, pixel clusters are nice though there might be a little refining here and there (a few instances of banding).
The head isn't there yet: it's more like a sphere + trunk than an elephant head. Check references to detect the basic features that make elephant heads unique. Pixeling of the trunk is not as good too. Drop all dithering, it can't work at this size and contrast. Take better advantage of 'natural' pixel line angles.
Colors are probably a bit too saturated (or too evenly saturated); also the dark tone might be a little darker, if only to stand out better from the PJ default grey.
My mouth is supposed to be open too, but I tried too hard to remove the vertical dark 2px line. The main thing was the pointy elephant lower lip.
Keep doing these for sure! Not necessarily 3 color though, it's mostly the small size that makes it a great exercise as you can really spend the necessary time on each single pixel.