It seems to me that game sizes are called different things in different groups, most notably with historical vs ahistorical. I'll try to list as many names as possible for each game size.
Skirmish/Squad Based/Squad on Squad/Squad Combat: Models act individually with very few exceptions. 15mm to 32mm.
Platoon/Skirmish: Models are individually based but mostly act together, each model will usually represent one person or vechicle.
Grand tactical/Epic scale/Small scale: Models are based together and one model can be multiple people or vechicles.
Dudes on a Map/Map Based: Models represent armies, manage whole campaigns.
Also, the following terms might need to be cleared up.
Narrative Game: Story is the focus.
Competitive Game: Balance is the focus.
Casual Game: Enjoyment is the focus.
Historical Game: Plausibility is the focus.
Wargame in a box: A board game with wargame mechanics, without the collecting and painting parts.