SuttaCentral
Quite a difference between Analysis of dhātus (Vipassanā) and Kasina (Samatha). But the difference between the Samatha breathing technique and Vipassanā Mindfulness of Breathing with some vagueness.
Something clear:
Any more clarity, if possible?
Hi! The basic idea in the distinction you mentioned is based on a later commentarial Abhidhamma view. There, they make a fundamental distinction between ‘concepts’ and ‘ultimate realities.’ There is some wisdom and perspective to be found in those teachings that can be helpful to certain people under certain circumstances, but historically speaking, the Buddha never taught that distinction and he didn’t teach mindfulness of breathing that way. So many on this forum will not feel particularly compelled to adhere to those distinctions where they aren’t necessary.
The dominant idea for Buddhists who are inspired by early Buddhism is this. If the Buddha really was the best of teachers (‘anuttaro purisadammasārathi’), and if his teaching really was well explained by him (‘svākkhāto bhagavatā dhammo), then later explanations that potentially conflict with his own explanations are not needed to practice at best. Otherwise he would have taught and explained them.
If you’re interested in what the Buddha himself taught, I recommend a study of the 16-phase mindfulness of breathing in the suttas. MN 118 is one sutta, and there’s also a collection of suttas on the topic at SN 54. There are reputable teachers with experience in the interpretation and actual practice of those subjects based off of the early teachings.
Wishing you the best in your journey and practice! 
The best clarity I have found on this topic is writings of Bhante Sujato - Books such as “A swift pair of messengers” & “The history of mindfulness" really clarified what the Buddha actually taught about Samatha, Vipassana, Jhana and all these topics that can easily lead to doubt when not well examined. I can’t recommend a better source!
I also came here to plug Bhante Sujāto’s book.
You can read it online or download in various formats in the top right
My apologies for horning in on this thread, but I have found myself stuck in the third tetrad of ānāpānasati for longer than I care to remember. I was told that there is a qualitative shift of focus upon entering that fourth tetrad, something along the lines of just this samatha-vipassanā dichotomy you’ve all mentioned.
I never really bought into it, but never came across a solid, sutta-based alternative teaching that I could turn to instead. So, I’d be open to suggestions of the EBT sort alluded to above. Thank you.
Hello, knotty! 
I’d recommend teachings from Ajahn Brahm and Ajahn Brahmāli on the Ānāpānassati Sutta. They hold generally the same interpretation (the latter being a disciple of the former), but they are also independent thinkers and practitioners. Also, the book linked above by Bhante Sujato for a more general overview that can be tied into ānāpānassati specifically.
Ajahn Brahmāli would give you a more technical, text-based analysis. Ajahn Brahm would give you a different flavor more matter-of-factly and experiential with similes and the like. Some of the things you’d find in those interpretations would be like:
As to the third tetrad, that is about developing samādhi and jhāna (as cittānupassanā primarily is). The final phase there (which says they train to ‘vimocayaṁ cittaṁ’—‘liberating the mind’) is a reference to ‘cetovimutti’ (liberation of mind), a term referring to jhāna states. That follows the step explicitly mentioning samādhi.
The suttas say that someone who experiences samādhi will naturally know and see the truth without needing to make an intention or act of will (c.f. AN 11.2). Just that is cause for reflection. Then, the last factor of the eightfold path is sammāsamādhi, defined as the four jhānas. The suttas explain that right samādhi will produce the resultant right knowledge and liberation, which are not included as path factors but as natural results of the path.
So one thing to keep in mind would be that something can be known by its effects. If someone says they have a pear tree that is oddly growing apples, I’d question if they really had a pear tree there.
Another example would be to imagine that someone in Plato’s allegory of the cave, who has never been above ground or seen daylight. If they went above ground for the first time, they wouldn’t need to make an act of will “may I be amazed and discover new things!” The sheer power of their new experience would contain all the data they need to just naturally inquire into the new amazing things.
Likewise, if someone says they have reached jhāna (culmination of the third tetrad) but are not finding insight, it would be quite questionable if they really had finished the third quarter, assuming they had learned about impermanence, four noble truths, etc. and were interested in those topics. The other possibility would be someone with wrong view who views those states as Brahman, God, and so on, but that’s very unlikely for a well educated Buddhist practitioner to blindly assume. If it isn’t coming naturally, the previous tetrads almost certainly need more development. That’s also where different views of jhāna can be put to the test in a sense. If it’s the real thing but doesn’t lead onward as the Buddha said it would, it’s very likely something before what the Buddha meant.
Otherwise, the Buddha’s simple instruction should be more than enough. ‘Contemplate impermanence.’ If it doesn’t automatically occur to the mind what to contemplate, just that should become very clear if the mind has been made ready. Like someone coming up from the cave being told “just look around you” to experience the daylight and objects there. Or someone who doesn’t know what a pear is just being told not to look at the trunk but look by the branches for fruit.
All that said, I apologize if it wasn’t really what you were asking for!
Wishing you all the best! 
No, I think you got it, right here…
It’s as if one were to say, “There is no fourth tetrad, only a third tetrad which extends into forever, with realization of impermanence, etc. arising sans ‘contemplation’”: which, oddly enough, appears to me to be precisely the solution to the samatha-vipassanā false dichotomy!
If that makes any sense.
Thank you!