In the glass mesh, the normals (direction it's facing) of the faces are not all pointing out to the air as they should.
So the fluid simulator does not know how to calculate the shape of the glass obstacle correctly.
To fix (EDIT) your second try:
With the glass selected,
Go into edit mode. (tab)
select everything. (a key)
Then "recalculate normals". (ctrl+n This is also in the tool panel at the left, in the "Shading / UVs" tab.)
You could make the domain object much smaller for more efficient
simulation. So you can resize it to just where the fluid will be.
Now you can turn up the Domain's fluid resolution...
You might want to make the glass a little thicker towards the bottom,
so the fluid does not stick through.
(EDIT. Noticed my instructions were just for your second try.)
To fix your first try:
On both the glass and fluid shape:
(In object mode, look at the scale of the objects, they're negative numbers, they need to be positive for the fluid sim to figure out the shapes.)
Apply the scale. (Ctl+A then click scale) Applying the scale makes the scale 1,1,1.)
Then recalculate normals. (If you recalculate the normals when the object is inverted, it looks ok, but it confuses the fluid sim.
(Also I didn't test if you needed to, but I'd also remove doubles on your meshes, the fluid shape has quite a few.)