I think that most of your post is on the right track. Even in the portion above, you identified the correct provision (18 USC 1960). Indeed, you've identified the provision that proscribes unlicensed money transmitting. But you haven't connected the definition of "currency" to that provision, or even identified why that that definition is relevant. Let's take it step by step and we can get to an answer 
The prohibition of unlicensed money transmitting businesses is 18 USC § 1960. This section makes reference to the registration requirements of 31 USC § 5330. The definition of "money transmitting business" therein makes reference to the reporting requirements of 31 USC § 5313.
The MSB regulations are in 31 CFR 1010. This part cites 31 USC 5311 et seq (among other sections) as its statutory authority.
Yes, this is all fairly convoluted. It's certainly possible that I've missed something.