Hockey is the name of the sport at the Olympics; introduced 1908 but permanently added to Summer games in 1924 (FIH, founded in Paris in 1924, now(?) 127 members). Much of its formation can be credited to English "public" schools.

Ice hockey was introduced to the Olympics in 1920 and became part of the Winter program in 1924 (IIHF, founded Paris 1908 as LIHG, now 52 members).
The need to separate the games with modifiers has grown since that time, especially in those countries strongly supportive of the game on ice (Russia, U. S., Canada, etc.) where "hockey" is understood by most people to mean ice hockey.