Did the Buddha according to the suttas mean anything else by the word "self" beyond simply the aspect of control/power? It looks like according to [the Anatta-lakkhana Sutta][1] one aspect that is considered "self" by the Buddha is power or control. Why did he use the word "self" instead of just "complete control/power"? Was/is there more to it than just power/control?   

Was there anything more meant by the Buddha according to the suttas by the usage of the word "self"?
This is a very famous line of argumentation in Buddhism, especially in Theravada, about no-self. The basic idea is that according to common sense, we can control ourselves by the power of will - move our limbs etc. - but we can't control external objects. From this it follows that this notion of control can be used to help clarify the boundary between "me" and "not me". Buddha then uses reductio ad absurdum to show that, in fact, according to this definition, nothing is truly in our control, therefore nothing is me (or "self").

Mahayana builds up on this basic point to show a deeper meaning the Buddha implied. Since nothing is fully in our control and things are always in flux (including our mind), said the Buddha, let's stop expecting that things can ever be perfect in some imaginary state of (conditional) Nirvana. This very expectation of perfection itself, says Buddha, is what creates the mental mismatch that we experience as psychological suffering.

In fact, our definition of perfection that we have internalized, our idea of how things should be, is exactly the core of our sense of self. This mental point of reference is what we measure the world against, and as long as we measure it, we will find things to be imperfect, causing the painful feeling of wrongness, dukkha. Only by doing away with any sense of territory, any idea of how things should be, which is the core of Self, can we open to seeing things as they are.

Therefore Buddha says, do not identify with anything, physical or mental. Identification leads to dukkha. First, because it (=whatever we identify with, e.g. body) will eventually break up and it's not in our power to stop that. Second, because it (=all external things, but also even the mind) is in flux and never fully in our control, therefore fundamentaly unreliable; impossible to guarantee. Third, because it (a hard position or view) leads to conflicts, arguments, and even violence.  And fourth, because it (i.e. an idea of how things should be) serves as a basis of comparison and setting of expectations, which is inherently conducive to dissatisfaction.

Therefore, a wise man, a rational man, a practical man seeking to stop dukkha, would be right to conclude that identifying with anything, any entity or concept, is simply a bad move, because it would inevitably lead to suffering.

Cessation of identification leads to cessation of dukkha. Realizing this, letting go of identification, and achieving the unconditional suchness (perfection without comparison) is the solution that Buddha offers. 

This is the meaning of this passage, in my understanding.