頭 and 顔: the head and the face
Use common actions to distinguish the head as a whole from the face.
The head
頭, read あたま, means head. 頭に帽子をかぶりました。 says that someone put a hat on their head; 頭に gives the location.
頭を下げました。 says that someone lowered their head. This could accompany a greeting or apology, but context is needed to determine why.
The face
顔, read かお, means face. 朝、顔を洗います。 says that someone washes their face in the morning. Do not change this into washing their hair.
顔を上げてください。 asks someone to raise their face. A request does not report that the action has already happened.
Movement and expression
頭を動かさずに、目だけ動かしました。 says that someone moved only their eyes without moving their head. だけ limits the body part that moved.
困った顔をしていました。 describes someone looking troubled. 顔 here helps describe the expression visible on the face.
Does 朝、顔を洗います mean washing one's hair?
Show the answer and why
No. It means washing one's face. Distinguish 顔 from 髪, which means hair.