It's apparent that different people have different notions of what a "native speaker" is. To a linguist, the term generally implies that a speaker has "internalised" the language through "natural acquisition", rather than through deliberate instruction/learning.
A "native speaker", as opposed to an extremely proficient second language speaker, can often make instant judgements about whether sentences "on the fringe" of the language's grammar sound grammatical. So for example, native speakers can probably instantly make judgments about whether the following sentences of English "sound normal":
Which students did you think had done their homework?
Which students did you wonder whether would turn up late?
These are the parents affected by the measures.
These are the parents baked a cake by their children.
It appears that a non-native speaker, even an extremely proficient one, will tend to make a judgement about these sentences much less readily.
There are other, essentially non-linguistic, definitions of "native speaker", e.g. "the language that I speak most and have the most cultural attachment to" or "The language that I acquired first". An issue which I'm actually currently discussing on another forum with fellow translators is that there are people claiming to be "native speakers" of English who write sentences such as "I have experience of translator since 4 years". I personally think this is an unuseful definition of "native speaker", but it shows how much confusion/variability there is.