There is no solid definition of the term of the sort that you're suspecting there might be: there is no agreement across RPGs for what exactly Thaumaturgy is or means, though outside RPGs it does have a solid (though pedestrian) English definition.
As KRyan and Simon Gill's answers have pointed out in excellent detail, the word itself, prior to ever being used in a roleplaying game, simply means "magic" without specifying any particular sort of magic. In the past it may have had denotations of miraculous workings or magic-from-a-machine, but modern usage of the word (such as it is) has faded to the point that it means merely "magic".
Various RPGs do use it to mean specific types or methods of magic, simply because it's a convenient English synonym for "magic" lying around frequently unused – a game designer that has already given game-mechanical meanings to more common words like "magic" and "sorcery" but finds they need another word for yet another type of magic-working, often finds "thaumaturgy" suitable.
You have noticed a pattern in its use, but that has more to do with it sounding "technical" and "latin-y" (actually Greek) than anything else – as good designers know, choosing a name that has the right "feeling" to it is an important tool for conveying the style of a game's setting and mechanics simultaneously. So, thaumaturgy often gets re-purposed in RPGs for technical, machine-based, ritual-based, or other complicated magic-working methods that our modern sensibilities would think of as being vaguely scientific, or at least rational.
All that said, the word as-is has no encumbrance of meaning beyond meaning "magic", so you can define it in game-mechanical and setting terms however you see fit without worrying about whether it's correct. Your proposed use for magic that is channeled through machinery is perfectly in line with common ways that RPGs have defined it, since it follows the common native English–speaking instinct that it sounds vaguely technical.