The lysate is after you have lysed your cells/homogenized your tissue. The input indicates what you are putting into some sort of column and the eluate is what you are eluting out of the column at the end of the protocol. I assume you are trying to purify a tagged protein?
Whole cell extract is what you have after you lyse your cells (which could be by bead beating, French pressing etc). The input is the clarified cell lysate (while cell lysate after centrifuging) that you add to your purification column. The eluant is what you obtain after washing the column and eluting target protein.
Cell lysates= when you done induction and expression after that you have lyse the cells with Tris, Nacl and glycerol. The lysed cells are called cell lysates.
After the lysed cells you have proceed for chromatography or Western blot. The amount of lysed cells you have used for chromatography or Western is the Input sample.
At last you get some amount of your desired product is known as Elution or eluted samples.