Did the Buddha according to the suttas mean anything else by the word "self" beyond simply the aspect of control/power?

No Buddha didn't.

Self and soul are common words. If Buddha didn't use social norms to explain Nirvana he'll say "attain Nirvana". But would we understand the whole of Dhamma with just those words?

I have asked a somewhat similar question before here.. Following the giant greek philosopher Parmenides. He said

The only roads of inquiry there are to think of: one, that it is and that it is not possible for it not to be, this is the path of persuasion (for truth is its companion); the other, that it is not and that it must not be — this I say to you is a path wholly unknowable.

So, it does seem futile to even think about no-self if it is not... in short, when we say the five aggregates are not-self it seems that we are saying the five aggregates are not unicorns.

But it is not so, Please read the comparative Analysis of Parmenides and Naagaarjuna here.

Naagaarjuna's response to your questions or to Parmenides objection of learning what is not is as follow:

Impurity cannot exist without depending on purity so that we explain purity by impurity. Therefore purity by itself cannot be attained. Purity cannot exist without depending on impurity, so that we explain impurity by purity. Therefore impurity cannot exist by itself. (Nakamura 61)

It looks like according to the Anatta-lakkhana Sutta one aspect that is considered "self" by the Buddha is power or control.

In my reading, the Buddha here was commenting on the subject of "self" rather than the subject of "control". He was not saying "control" is "self". He was saying if there was a "real self", this self could control itself.

If the claim of "self" being an illusion is made, one needs to at least know and preferably define **what "self" is.**

"Self" is defined in SN 22.81 is an "assumption", "mental formation" & "ignorant misunderstanding".

There is some other explanation to this that I am currently unaware of.

"Self" is an underlying tendency (anusaya). Using scientific terminology, it can be said "self" is a survival instinct (i.e., a form of craving called craving to be). When the mind becomes anxious or lustful, these energies of anxiety & lust create ideas or thoughts in the mind of 'self' or 'identity'.

Thus Buddha taught the arising of 'self-view' is the 'arising of suffering' (SN 12.15; SN 5.10).

Why now do you assume 'a being'? Mara, have you grasped a view? This is a heap of sheer constructions: Here no being is found.

Just as, with an assemblage of parts, The word 'chariot' is used, So, when the aggregates are present, There's the convention 'a being.'

It's only suffering that comes to be, Suffering that stands and falls away. Nothing but suffering comes to be, Nothing but suffering ceases.

SN 5.10

Option 3) There is something you have not thought of.

Your claim 'if one sees an illusion of a lake it requires the knowledge of an actual lake which requires an actual lake to exist.' is not true. It's easy to see it as true when you are choosing, as an example, something that is concrete and obviously conventionally true.

Very often, when people visualize things, either awake or dreaming, it is with the idea that they have a perfect vivid image in their mind of what it is they are thinking of. However, you can test this by asking a group of people to close their eyes and visualize something, then ask them to draw it. You will find that people often draw things very badly even if they thought, a second ago, that they had a picture realistic view in their head. You might be inclined to say that this is due to lack of artistic talent, but I can tell you from experience-- you can get a classroom of artists to draw something sitting on a table in front of them and get a pretty good sketch from each of them, but you'll find MUCH more varying results from asking them to draw from their imagination, even if they all insist that their internal image is photorealistic.

In reality, people very rarely (if ever) actually have a very detailed image in their mind. What occurs is that they have PART of an image, and the rest of the details are vague. They may know exactly how 3 parts of it look up close, and how the colors should vaguely be, but not how anything is actually supposed to fit together. However, the way the mental image works, it's hard for the mind to actually focus on the parts they don't know and especially hard for the mind to notice that those parts are incorrect. Once you must act based on the information in those parts (via drawing) suddenly it is hard for the mind to guide based on this bad image. Without that act, drawing, it is very difficult to realize what parts of the mental image are disproportioned or incorrect, or even that any of it is.

That may have seemed like a long digression, because, like the lake, you still are basing the image on something that physically exists.

But the same thing occurs when you ask people to draw abstract images that they visualize beforehand. When the only goal is to draw an image you have imagined, you still find that you have failed to imagine it to the detail you thought you had. It's less noticeable and, from an art perspective, it's much easier to still produce an art piece that looks very good, despite not being what the artist was visualizing to begin with.

It is like that for concepts as well. People begin to think of a concept-- say, self-- and they see some very zoomed in details. They learn these details from different people, and when you ask people, "what is the self" they mentally create something like the previously mentioned vague visual image- they might zoom in on details, they may create a vague shape. Much like the abstract visuals people made previously, when they must act on this, they might find that it's impossible and self contradictory. Usually, like the artists, an uninstructed person just patches it up and makes something different as they go.

The doctrine of not-self is the Buddha taking all those zoomed in details of what people often try to put into the 'self' concept and saying that they don't fit together into this true concept of 'self'. He was responding to how other people used words, and he was specifically saying that these patched together collages of details did not work together like they should.

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.