What does these phrases mean?

"on the east", " in the east" and " to the east" is relative to the point where you are when you say it.
Example : China and America

1) if you're in America,
you say : China is to the east of America, not on or in b'coz that would means China is located in/on the east of America itself.

2) on vs in
If you 're in China
you say : China is in the east and located on the east side of the world

2a) if you're in America,
you can only say : China is on the east side of the world.

see the difference?

"On the east" is incorrect. You could say "on the eastern border", which says that it is at the border which is in the eastern part of a country or something.

"To the east" means that the compass direction from one to the other is east. "In the east" would be used more to describe a portion of a country rather than a relation between two countries or states. I could say "cities are closer together in the east" when talking about the US. That means that in the eastern portion of the country, cities are closer together than in other portions of the country.

So... "in the east" = describing the eastern region of some country/state/city. Talking about one entity.
"to the east" = describes the direction from one location to another. A relationship between two entities.
"on the east" = incorrect. Not used.

what u say about to the east does not seem correct. anyway, in the east seems to me to be used
when mean in the eastern region/area, e.g., in the orient/asia or in the eastern part of the u.s., whereas the other to (on the east, to the east, and also east of) refer to relative positions or directions. now let's see how to use these three. we say palm spring is to the east of los angeles or east of los angeles, you cannot say on the east of. and that's the main difference i that: on the east cannot be followed by of X or anything equivalent because on the east already means to the east of the place being discussed, to the east of here/there.