I believe you may be slightly confused about terminology used. It is really pretty simple.
An internal symmetry is indeed a symmetry of the theory that does not impact the spacetime coordinates. It can be a global symmetry or a local symmetry.
Examples of a global symmetry are $U(1)$ for the EM charge, $SU(3)$ flavour or $SU(2)$ isospin.
When the internal symmetry is local it is a gauge theory. The gauge fields are then a connections of the symmetry. Examples are $U(1)$ for the EM force, or $SU(2)\times U(1)$ for the electroweak force or $SU(3)$-color for QCD. Note that gravity has a local symmetry as well: diffeomorphism invariance. It is a gauge theory, but not an internal symmetry.
In general the confusing about this comes from:
traditionally gauge theories are taught in the following way: one starts from a global internal symmetry, usually $U(1)$, and then asks oneself the question: what happens if we make this local. One then see that one has to introduce a connection etc. There is a prior no reason to do it that way; it is just educational convenience/laziness
a local internal symmetry is, strictly speaking, not a symmetry. It is a redundancy in the description of the system that allows to simplify it. It introduces d.o.f. that then need to be gauged away by fixing the gauge.
the example of gravity that is a gauge theory but not an internal symmetry
the fact that some people use gauge theory and Yang-Mills theory interchangeably. They are not the same. YM are just special examples of gauge theories.