Even longtime collectors often get this wrong or continue to use the nomenclature incorrectly just because they are used to one terminology over another.
HIGH POWER: The original FN terminology as marketed in all languages, applies to all pistols marked Fabrique Nationale. Including prewar, wartime and postwar manufacture. HIGH POWER was, and still is, the correct FN factory terminology.
HI-POWER: This term was introduced by Browning USA (in the 1950s) for their commercial pistols when they planned on importing the (Mauser based) sporting rifle. The rifle became known as the HIGH POWER RIFLE. The pistol became the HI-POWER. This terminology applies only to pistols marked BROWNING ARMS COMPANY. The majority of these pistols were commercial sales in the U.S. and Canada. The term was also used later by Browning Intl. after FN acquired Browning.
To summarize: If the pistol has a FABRIQUE NATIONALE slide legend, it is a HIGH-POWER. If it is BROWNING ARMS COMPANY marked, than you have a HI-POWER.
Many collectors may not care about this little nuance. If you think of a wartime pistol and keep describing it by a 1950s term… maybe than does the historical accuracy gain importance.