Is there a real difference between "not-self" and "no self", and if so, which one is correct?

Whichever way you say it 'no-self' or 'non-self' the wisdom from the epoch of Greek philosophy will ask you what is that self you predicate while claiming its non-existance?
Parmenides will rear out his head saying:

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.

Me think only those who know the end which "is" can know the meaning of anattā

The way I understand the Pali term "Anatta" is translated as no-self or notself by some. There is a great debate about whether Anatta means notself or no-self. The only way you can get an answer to this question is only by realising Anatta or at least by eliminating Sakkaya Ditthi. (personality belief)

As I view this concept: "Not self", you are focusing on things other than "self", thus you are assuming "self" exists.

However, "no self" is something else. You conceptually recognize that "self" is fake. And you escaped from the "self" cage. Thus "no self" or "'Keine' self", you set the "self" variable as 0.

Hope it helps!

No-Self is one extreme.

Quoted below is from Chandrakirti Commentary.

so too, the buddhas have taught, “There is neither the Self nor the Selfless at all.”

Just as the view that there is a Self is not ultimately true (atattva), so too its opposite—the view that there is no Self—is also not ultimately true. In this way, they taught that there is neither any Self, nor any non-Self whatsoever. This is as it says in the Āryaratnakūṭa:

Kāśyapa, “Self” is one extreme, and “Selflessness” is a second extreme. The middle that is between these two extremes is cannot be indicated (arūpya), it cannot be shown. It has neither location nor appearance. It cannot be represented, it cannot be marked. Kāśyapa, it is called the middle way, the true discernment of things.

Some thinkers such as the Sāṃkhyas perceive that, in the case that karmic imprints (saṃskāra) go out of existence every moment, there is no relation between karmic acts and their results. Perceiving this, these thinkers posit a Self [so as to provide some basis for that relation].

The Lokāyatas, on the other hand, through their reasoning see no-Self that would be an agent of continuity [from one life to the next]. Hence, they assert that there is no Self. They do so with statements such as, There is a person just to the extent that there is an object of the senses. Oh Bhadrā, this is what the learned say, and it is a jackal’s foot.

But just as those without cataracts do not at all see the hairs, flies and so on that are perceived by those with cataracts, so too the buddhas do not at all see any real entity (vastusvarūpa) such as a “self,” or a “non-self” that is imagined by childish persons. Not seeing such things, .. the buddhas have taught, “There is neither the Self nor the Selfless at all.”

The statement sabbe dhamma anatta (Dhp 279) means all phenomena is not self or everything is not self. A phenomena or thing (dhamma) refers to anything - the five aggregates, the body, the mind, nibbana, the teachings, the Buddha, thoughts etc.

It also means that there is no self in all phenomena or that there is no self in all things. We can also say that all phenomena is empty of a self - from Empty Sutta.

This means that when you dig deeper into anything, you won't be able to find anything like a self.

But this doesn't mean that there is no self at all, as the Buddha himself said in the Self-Doer Sutta. The self is a thought in the mind that emerges when the five aggregates work together, according to dependent origination. But if you break down the five aggregates into its constituent parts, you cannot find the self anywhere.

This is similar to how a lute (a musical instrument) works. When different parts of the lute work together, they produce music. But if you break the lute down to its constituent components, you cannot find music. This is taught in the Lute Sutta.

That the self is a mental idea, a thought in the mind, is taught in the Quickly Sutta referring to the "I am the thinker" mental idea.

The concept is the unreality (Paññatti).

The unreality neither arise&vanish (Saṅkhata;Khandha) nor be the opposite of arising&vanishing (Asaṅkhata;Ṅibbāna).

The Self (attā) is a subset of the concept. It is an unreality.

the no-self, non-self is the opposite of concept. Saṅkhata and Asaṅkhata are the opposite of Attā-Paññatti (self).

The concept is imagined by thinking of the reality. There are 2 type of concepts:

1. Name-let people know something.
2. Imagination-imagined by people.

The imagination has many various subset such as home, time, direction, tall, big, etc.

Self is in a set called illusion of compactness (Ghana-Paññatti).

Another, someone may confused by Anattā in AnattaLakkhaṇaSutta, then say Nibbāna and Pañnatti is Attā, or neither attā nor anattā. That is just logic, mistaken logic exactly. AnattaLakkhaṇaSutta is for unloosing from Khandha, not for describing everything. As I describe above that if you describe Nibbāna as attā, nibbāna will be unreality. and if you describe Nibbāna as neither attā nor anattā, nibbāna will be neither reality nor unreality. **Then people who already enlighten Nibbāna must completely disagree your conclusion surely.

The difference is significant.

To say: "This is not the self", is to point to some existing thing and having examined it with the criteria for what is worthy to consider the self (that is, that it is something that is under one's control) and finding it out of one's control, the statement is simply the expression of an observable, provable fact.

To say: "There is no self." is to express an opinion (point of view, diṭṭhi) as to all things throughout time and space, past, future, present. This is something that is beyond possibility, as even for Buddhas, for 'Incalculable is the beginning, brethren, of this faring on."

SN 2.15.9 http://buddhadust.net/dhamma-vinaya/pts/sn/02_nv/sn02.15.009.rhyc.pts.htm#p1

In other words, if it cannot be seen and proved, it is just an opinion.

The Dhamma is clear on the practice: opinions are something to be let go; identification with that which is not the self is something to be given up.