  We present an explanation of how the $\mu$-deformed Segal-Bargmann spaces,
that are studied in various articles of the author in collaboration with
Angulo, Echevarria and Pita, can be viewed as deserving their name, that is,
how they should be considered as a part of Segal-Bargmann analysis. This
explanation relates the $\mu$-deformed Segal-Bargmann transforms to the
generalized Segal-Bargmann transforms introduced by B. Hall using heat kernel
analysis. All the versions of the $\mu$-deformed Segal-Bargmann transform can
be understood as Hall type transforms. In particular, we define a
$\mu$-deformation of Hall's "Version C" generalized Segal-Bargmann transform
which is then shown to be a $\mu$-deformed convolution with a $\mu$-deformed
heat kernel followed by analytic continuation. Our results are generalizations
and analogues of the results of Hall.
