I'm not a psychologist, so I am only familar with the other definitions of psychic, ie. having to do with the supernatural, paranormal, etc. It's been discussed in the forum before: the first definition listed in the dictionary is not necessarily the most common sense of a word. In fact, it's often the case that the first definition is the least
common sense in use. That's true with Webster's; I'm not as familiar with other dictionaries, but it seems to be the case here with Random House as well.
Again, as a layman, I would understand "psychic problems" in a medical context as "problems with the psyche". I have at least a vague notion of what the "psyche" is, but I don't think that is the same thing as having "psychological problems". Maybe an expert can explain to us what the difference is. I can't tell whether the LEO entry is wrong or not (there are several entries containing "psychic" in your link), but I do think the entries need to be tagged as specialized terms, ie. psychol. or medical or whatever. I don't think this sense is known to laymen.
Edith: Here's the M-W entry for reference.
Main Entry: psychic
Variant: also psychical
1 : of or relating to the psyche : psychogenic
2 : lying outside the sphere of physical science or knowledge : immaterial, moral, or spiritual in origin or force
3 : sensitive to nonphysical or supernatural forces and influences : marked by extraordinary or mysterious sensitivity, perception, or understanding