I can't easily tell a difference between "no self" and "not self" in English -- and if there is a difference (if you can construct a distinction in meaning) I think it's too fine a difference to be significant -- like if someone tells you something is "a mile away" and you get out your micrometer.
I mean, the "a" prefix in Pali is a negator, which appears on all sorts of words:
- kalika ("time"), akalika ... how do you translate akalika? I say "timeless" because I think that happens to be a felicitous word in English, but equally be "without time, "outside time", "not subject to time" or "not controlled by time" or "nothing to do with time"
- verena ("hatred"), averena ... how do you translate averena? I say ""non-hatred" (because I hope that's the most literal translation even if it isn't idiomatic English) but some translations say "loving-kindness" -- i.e. not just an absence of hatred, they substitute one of its "polar opposites"
Maybe "selfless" could be a translation, however in English that's an adjective used to describe "altruistic" behaviour (so that's not an appropriate translation).
anatta is grammatically a noun. I don't, I wouldn't, normally equate two nouns -- "a dog is a cat" for example doesn't make sense. So I usually read anatta as if it were an adjective ("a dog is hairy"), i.e. as if "form is anatta" is describing a property or characteristic of form. And lakkhaṇa
(translated "characteristic") in Anattalakkhaṇasutta suggests that seeing anatta as a characteristic -- an adjective not a noun -- isn't bad.
One time is it appropriate to equate two nouns is when you're talking about categories -- "a dog is an animal". Perhaps anatta can be seen as a category of things, especially in phrases like Sabbe sankhara anatta.
I think of "self" as a verb in that kind of context ...
Calvin and Hobbes - Verbing Weirds Language - by Bill Watterson for January 25, 1993
... i.e. "selfing" is an action, it's something you do to things -- you "take as 'self'" or "perceive as self" or "attach to as if it were self" -- and an anatta thing is something which you shouldn't "self" in that way, something which isn't fit to be selved.
Perhaps you can see that the Anattalakkhaṇasutta is about the five aggregates. The Dhammacakkappavattanasutta mentions aggregates too:
Rebirth is suffering; old age is suffering; illness is suffering; death is suffering; association with the disliked is suffering; separation from the liked is suffering; not getting what you wish for is suffering. In brief, the five grasping aggregates are suffering.
jātipi dukkhā, jarāpi dukkhā, byādhipi dukkho, maraṇampi dukkhaṃ, appiyehi sampayogo dukkho, piyehi vippayogo dukkho, yampicchaṃ na labhati tampi dukkhaṃ—saṃkhittena pañcupādānakkhandhā dukkhā.
So I think of "selfing" as "grasping" or "attaching" (especially to aggregates).
I think that colloquially the word atta had two meanings:
I think that informs the meaning too -- to say that the aggregates are anatta is to say that they shouldn't be regarded as "soul" or "permanent self", which fits in with such things being anicca (impermanent).
Since you're asking I don't think it goes so far as to say "there is no soul" or "there is no self" -- instead see e.g. How is it wrong to believe that a self exists, or that it doesn't? -- that would belong to the "thicket of wrong views" cited here.