These are illustrations of the equation for the general sinusoidal function and the simple harmonic motion (SHM):
I am having confusion about the way the equation for the general sinusoidal function is written with a minus in front of the phase shift while the SHM function has a plus sign and the periodicity factor is outside the parenthesis while the angular velocity (SHM) is inside the parenthesis.
In the sinusoidal, you have to take the periodicity factor out of the parenthesis to have the real phase shift while in the SHM you don't have to.
What is the distinction between the two functions and why are they written differently?
Your first equation describes a sine function plus a constant. If $k\ne0$ there is no way to find a relation between the parameters of the two function because they are different. On the other hand if $k=0$ the relation is trivial: $$ y\mapsto x, x\mapsto t, a\mapsto A, b\mapsto\omega,h\mapsto -\phi/\omega. $$
Thus, for $k=0$ the functions are equivalent up to definition of parameters.