How does Nyquist bit rate formula relate Bandwidth, Signal Rate, and Bit Rate?
You mention that you are getting stuck on they Nyquist bit rate formula. I think the sticking point for you is that the Nyquist formula only gives the upper bound for bit rate, for a given bandwidth. In other words, the Nyquist formula doesn't state that \$BW=S\$, but it could be interpreted to mean that \$BW \geq S\$.

A few other (hopefully) helpful bits:


Signal rate \$S\$ is not necessarily equal to bandwidth. There are a number of different modulation schemes or even line level codings which could all have the same symbol rate but have different bandwidth (i.e. occupy a wider or narrower range of frequencies).
Similarly a symbol (related to signal rate) can encode any number of individual bits (even a non-integer number in some fancier schemes), so signal rate is not necessarily equal to bit rate either.
Honestly I'm not sure where \$c\$ comes from; I haven't seen that particular formula before. What's the source? If you look at this calculator you will see that their formula only cares about data rate, bits per symbol, and forward error correction (which we're not using here so presume that FEC=1), so there is no \$c\$ factor to be found.


UPDATE

A better treatment of the relationship between symbol rate (signal rate) and bandwidth can be found in this paper. Relevant excerpt:


  The maximum feasible symbol rate and the bandwidth of the communication medium are directly proportional,
  Symbol rate = α · Bandwidth
  with a constant of proportionality α ≤ 2 (in practice, α ≈ 1.8).