Suppose I have a sinusoidal signal of the form $x(t) = a\sin\left(2\pi b(t-c)\right) + d$. I understand $a$ is the amplitude, $b$ is the wave number, and $c$ is the phase shift. I know that the term $d$ shifts the entire signal vertically, but does it have a specific name in signal processing?
It is usually called DC offset or DC component. Since you use $t$ as a variable, which suggests "time", the quantity $b$ would normally be called frequency. Also note that $c$ is not the phase but a time offset. The phase $\phi$ is usually defined as in
Comparing (1) to your formula gives $\phi=-2\pi bc$.