To answer the technical content of your original question: While you write the proposal, technically it is submitted by your institution to the National Science Foundation. If it is funded, then the NSF gives the requested money (or any other amount the NSF decides on) to your university, which then pays it to you. So this corresponds to your option (b).
But let me also comment on a separate issue: The fact that you ask this question here suggests to me that you're trying to navigate NSF rules on your own, without letting anyone more experienced at your institution guide you. This is not a winning strategy. NSF rules are plentiful, and you're going to run afoul of many of them if you don't rely on the help and support of experienced grant writers and administrators at your university. There will be many things you don't know you don't know. You will also find it difficult to write successful proposals if you don't ask others with experience for help. Reach out to others at your institutions and see that they can walk you through things such as the question you asked, or the issue with the 2-months rule mentioned in comments on RLH's answer. You will be more successful that way in the long run!