You didn't read all of that, otherwise you wouof have noted that the word ordnance doesn't appear once.
Arms vs Ordnance - What's the difference? | WikiDiff
Hair splitting, yes...but this is a thread for that. Arms refers to military, offensive and defensive, and fighting weapons.
Ordnance can mean the same thing, but is most often used to refer to explosives. It even has the artillery in its definition, while arms does not.
Further, the people of 1776 and beyond were not permitted to own canons, the ordnance of the times. That had to be registered, and I believe it cost 200 dollars...which was a small fortune. Basically limiting ownership to privateers and well funded militias.