Is the 1922 peace silver dollar the same as the liberty silver dollar?

Technically a Peace dollar is a Liberty dollar because all silver dollars minted up till 1935 carried a picture of Miss Liberty, but not all Liberty dollars are Peace dollars - only those made with the specific PEACE design from late 1921 to 1935 are Peace dollars.

Normally each type of silver dollar is given a more specific name based on its design, e.g. "Seated Liberty", "Flowing Hair", or Morgan or Gobrecht for the artists who designed them.