What is the difference between logarithmic decay vs exponential decay? I am a little unclear on whether they are distinctly different or whether this is a 'square is a rectangle, but rectangle is not necessarily a square' type of relationship.

The "Square is a rectangle" relationship is an example where the square is a _special case_ of a rectangle.

"Exponential decay" gets its name because the functions used to model it are of the form $f(x)=Ae^{kx} +C$ where $A>0$ and $k<0$. (Other $k$'s above $0$ yield an increasing function, not a decaying one.)

Similarly for "logarithmic decay," it gets its name since its modeled with functions of the form $g(x)=A\ln(x)+C$ where $A<0$.

These two families of functions do not overlap, so neither is a special case of the other. The giveaway is that the functions with $\ln(x)$ aren't even defined on half the real line, whereas the exponential ones are defined everywhere.