'atha yogAnushAsanaM'.

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'atha yogAnushAsanaM'
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'The thinker is the thought.' could be interpreted in two ways. Could give us 2 meanings. the idea of the existence of a 'thinker' is a thought only, like other myriad thoughts. When thought says 'there is this table, then we know the meaning of 'table'. when 'The thinker is the thought.' is said , we have no such an object like 'table' there, which has been pointed out in this statement. So this simply means 'Thinker' is but a 'thought' moving like so many other relevant or irrelevant ,rational / irrational thoughts. There is another meaning as is suggested by you. I would explain this in this way : 'thought' is the functioning of 'brain' in any specific physical 'body'. And there is also a 'consciousness' which knows this fact. The 'consciousness' is not saying a word. It is simply there and the movement of thought happens according to its own rules. The 'memory' is a store-house' of all such verbal assertions, and 'a meaning' given to a 'thought' is just formal, or is again in terms of a 'thought' only. The question is : What is this 'one' who / that claims oneself as the 'thinker'? 'consciousness' does not nor can claim this. Obviously It is the association of two thoughts that gives rise to the apparent existence of 'a thinker', who is just not there. This false notion of emptiness assumes the form of 'a thinker' in memory. Though false, it is the centre of all creation that is within the periphery of 'thought'.
The whole phenomenon 'the thought + the thinker' presuppose a support and ground, where-in this play happens and there is a 'witness' which is self-evident and not an 'inference' like thought / thinker. And this word 'witness' is being used in a loose sense. Let us say, there is such an entity which could be termed as 'witness' and by no logic, experience or evidence of any kind could be denied to have 'existence', though we can't say anything about its nature and form. And this is never a 'person' of the kind. We can say it is 'impersonal'. We should not hurry to describe its way of functioning, but accept that this thing is there. If we deal with this thing through thought / logic, we shall stuck in the groove that is thought / thinker. We need a kind of 'silent watchfulness' only, that couldn't be 'practiced'. And we should say 'we don't know'. Regards.
And I sincerely feel Vedanta has a way to it, 'Right meditation' has also a way to it, but don't know if the traditional religion of east or west or of anywhere else could ever 'touch' it. Nietzsche said 'God' is dead. This appeals to me. The 'God' as 'thought' is dead the moment he is born as a thought. And we really need to say 'good-bye' to that God, if we want peace and happiness on earth.
'yoga' in line with all major vedik scriptures begins with the aphorisms 'atha yogAnushAsanaM' , 'yogashchittavRttinirodhaH' , 'tadA dRShTuH swarUpe'wasthAnaM' and 'vRittisArupyaM-itaratra'. This puts forward in clear words what is 'yoga'. This treatise by Patanjali talks about 'vRitti' which in all its implications could be taken just a Sanskrit word for 'thought'. And then further explains the nature and essence of thought, and how unless this thought is 'silenced' one can't see one's Real form.... And Both SAnkhya and Vedanta don't attempt 'defining' or explaining and mentioning 'God'. And that is how even an 'atheist' may sometimes find these two streams of SanAtana-dharma agreeable.
I love J.Krishnamurti who started with this 'phrase-like' statement 'Thought is the thinker'. 'Experience is the experiencer' .. This opens up an altogether different approach away from 'traditional ways of thought'. And though J.Krishnamurti hardly mentions about Vedanta and the related trends (and for the obvious reasons), he begins with the mention of 'consciousness' which is truly the connecting cord for all and every-one, and who-so-ever could find a thread could go further with him.
Q.:Does the brain think or is the brain a thought?
Brain is but a system where thoughts appear and disappear, emerge-out and subside again and again. As such every 'thought' is a wave-like movement that could not be captured in a concrete shape and form. And 'there is a thinker, and I am that thinker' is again a delusive thought only, that surfaces at times and stays as a habit in brain.
Sri Ramana says each and every thought is invariably associated in this conscious or sub-conscious thought 'I AM'. Patanjali also points out to the same, but indirectly. He talks about 'asmitA-samAdhi'. where-in one (aspirant) understands that 'I AM' sense is also a 'thought' only and is got rid of by this understanding.
Q. :Wonderful, all these thoughts from Africa, India, Europe.... but in the end we do not even know where they are from, what they mean and who is thinking or witnessing them.
A. :'...who is thinking or witnessing them...' But could the fact of witnessing(?) be denied or refuted,
Q. : Sounds a little bit like Descartes (cogito, ergo sum).
A. : There is one 'I am' that Descartes speaks of : 'I think therefore I am', There is another 'I am' Vedanta speaks of, :  'I am therefore I think'.
The 'I am' Descartes talks about is an 'inference', The 'I am' Vedanta talks about is the 'consciousness' devoid of sense of being a specific person, where 'The observer is the observed'. Though J.Krishnamurti says this keeping in view a very different meaning then Vedanta. And what J.Krishnamurti says applies to a person who is a seeker or not even a seeker., while what Vedanta says is applicable to both A jnAnI or an ajnAni (one who has realized the Brahman /Self / Atman, and  one who has not realized the Brahman / Self / Atman, respectively).