Taijiquan is the most well know internal art, but baguazhang and xingyiquan along with hundreds of qigong exercises (many of them abstracted from one of those three arts) have been steadily gaining in popularity. The following is meant to be helpful in understanding the term, 'internal.'
If you move your tongue around in your mouth and then do the same thing looking in the mirror it will appear that your tongue is moving differently than it feels. The tongue tends to exaggerate the size of objects it touches. This becomes really obvious when you have a cut on your tongue. Similarly, the back of the palms and the front of the wrists perceive heat and moisture quite differently. (You can try it right now.)
The internal organs move around like the tongue and each has it's own very specialized sensory and motor nerves, as well as its own intelligence. The feeling of lifting up your right kidney feels very different than the feeling of lifting up your right shoulder, but both can be felt. Rotating your liver feels very different than rotating your head, but both move independently. Our internal organs move around semi-consciously most of the time, completing specialized functions automatically.
This 'internal' movement necessarily supports all our other movements. This is experience is the basis for Structure school of Chinese medicine. The premise of which is that chronic illness, injury, "deficiency" or "excess" will have a physical impact on the underlying structure of our bodies. It will eventually reshape even our bones.
All our 'external' movements like waving our hands or wiggling our toes are interdependent with internal movements for support. This is part of the function of our organs, our vessels, glands etcâ€¦, form is inseparable from more obvious function( their form shape and movement have a function in addition there systemic functions). When that support is partial, inhibited or too abrupt we say qi flow is inhibited or restricted. Over time these qi restrictions may become imbalances, stiffness, collapsing, or pain, in both 'internal' and 'external' movement.
Qi gong teachers have many devices for developing students' sense of the internal. Remember that the concept that qi itself is not restricted to or limited by ideas of internal or external personal space, it's bigger than that. Working with the concept of qi means not restricting our view to just organs, or even the limits of the physical body, it would suggest an expansive view, and a softened focus. [Where you practice matters!]
Thus it follows that this seemingly infinite movement inwards also continues as our gestures, movements, and our senses move out into space. Tying the internal to our conduct and to the shape of the environment we live in.
Imagination is a necessary component of feeling. Most people feel their liver moving, they just havenâ€™t named it and thus, in not naming it, they have not differentiated it. (is it still part of undifferentiated chaos?) Feeling is a type of distinction which requires some imagination and some practice.
Therefore Iâ€™m dubious of distinctions between mental and physical.