In that case, I'd argue that it's actually fine that they just manifest the weapons in, since they're basically going "hey god, I could really use a weapon for my friend here, can you hook me up?"
It'd be weirder, in that situation, if they then had to make the weapons themselves instead of the gods they're entreating just manifesting them into the world.
In-universe, the reason for the "smith" part of "animasmith" was probably something like "they create special weapons" > "the people who normally create weapons (of the standard variety) are usually blacksmiths" > "the word 'blacksmith' is sometimes shortened to just 'smith'" > "we should call them 'animasmiths' (with the 'anima' part coming from either the fact that they have animal traits, or possibly due to souls having something to do with either the way they create things or the weapons themselves)".