I do think that website does make it more confusing heh.
If we go back to the early 18th century then we have lots of the smaller 3 and 4 pounders. They might have had a few dedicated gunners but the rest of the crew was drawn directly from the infantry battalion it was supporting. Such guns were IIRC drawn by horses but left to themselves on the battlefield and performed the role of direct support for its battalion.
Although you still see some of that in Napoleonic times it was not as widespread and some armies did not use it at all.
The Napoleonic foot artillery did not have any different armament for its crews that I know of.
The armies I know of simply used terms like foot and horse artillery but maybe the US army used the term mounted too in the early/mid 19th century.
ETW does have fixed guns in the beginning and I guess that is one way of making the technology change work in the campaign. Guns in the early 18th century was rather on the heavy side and with the use of civilian horse teams that ran away at the first shot that might be the reason why CA decided to make them fixed.
Edit: From your link it seems "mounted" was more of a term of honor to show the difference between those who were cross trained and those who were not. Their function still seemed to be the same as the regular foot artillery.
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