Personally I try to avoid to use Wikipedia by itself as a source of information, like an encyclopedia it is a good way to get quick facts but you still want to verify these facts with an outside source, However, since you yourself are using Wikipedia, I am just going to stick what they have to say on the matter:
"A "shoot 'em up", also known as a "shmup" or "STG", is a game in which the protagonist combats a large number of enemies by shooting at them while dodging their fire. The controlling player must rely primarily on reaction times to succeed. Beyond this, critics differ on exactly which design elements constitute a shoot 'em up. Some restrict the genre to games featuring some kind of craft, using fixed or scrolling movement. Others widen the scope to include games featuring such protagonists as robots or humans on foot, as well as including games featuring "on-rails" (or "into the screen") and "run and gun" movement. Mark Wolf restricts the definition to games featuring multiple antagonists ("'em" being short for "them"), calling games featuring one-on-one shooting "combat games". Formerly, critics described any game where the primary design element was shooting as a "shoot 'em up", but later shoot 'em ups became a specific, inward-looking genre based on design conventions established in those shooting games of the 1980s."
"Shoot 'em ups are a subgenre of shooter game, in turn a type of action game. These games are usually viewed from a top-down or side-view perspective, and players must use ranged weapons to take action at a distance. The player's avatar is typically a vehicle under constant attack. Thus, the player's goal is to shoot as quickly as possible anything that moves or threatens him. In some games, the player's character can withstand some damage; in others, a single hit will result in his destruction. The main skills required in shoot 'em ups are fast reactions and memorising enemy attack patterns. Some games feature overwhelming numbers of enemy projectiles and the player has to memorise their patterns to survive. These games belong to one of the fastest-paced video game genres."
The sections then continue to elaborate more on each of these points and even define different subgroups of shoot-em ups and different application of the terms. All of which follow the same logic overall (explained later).
Both of these had heavy level of citation (sometimes Wikipedia does do its job properly). However, they both share a very common thread, shoot-em ups are subgenre of shooter games and thus a subgroup of action games. The term is rarely used to describe actions and activities outside of that genre, as in shoot-em up is more of a category of games than a adjective someone can apply liberally wherever they want. This follows in line with the industry's accepted usages of the word by game developers and game media.
I am sorry, but you can not cherry pick a single sentence out of context of the broader definition and use it to apply your own meaning to what that sentence means. The whole definition matters, and the context of what the sentence means by shooting matters.
As I said, feel free to use the term personally to describe a certain style of gameplay as a personal or group slang. But again, your definition of shoot-em up is not the broadly accepted definition, and as such is not common usage.
As for your question what would be a better term for games that have high level of micromanagement of units where you dictate their attack (or 'shoot') orders individually. Give me some time as you can see I am very strict with the formal definition and usage of terms, which means I will have to verify the different terms used to describe such gameplay. One term I did find was unit based tactics or unit based tactical combat (as in you individual command each unit separately and completely).