The Buddha did indeed teach the noting technique. For example, in the Pali text of the Satipatthana Sutta it says:
"Gacchanto vā, 'Gacchāmi't pajanāti. When walking, he knows, "I am walking." (or simply "When walking, he knows, 'Walking'.") Likewise, When a lustful mind is present, he know, "The mind is lustful." etc.
The noting is the mental factor of jhāna called vitakka, which applies the mind to the object being noted. Without it, the mind will just wander here and there at random. In the second jhāna stage, the initial application will no longer be needed as attention will be sustained on the object currently being observed. Nevertheless, it is never really dropped, as the mind won't remain in deep concentration forever. Whenever attentiveness begins to slacken, one needs to resume noting, or if secondary objects distract the mind from the primary objects, the secondary objects need to be noted as "hearing, hearing," or "thinking, thinking," and so on.
To explain the importance of noting in detail will take too long. If you read the whole of "In This Very Life" with great care and patience, the vital importance of noting everything in detail will become clear, but only if you practice intensively in conjunction with reading or listening to teachings. Theoretical knowledge and practical experience are poles apart.
might be a good place to start.
• • • • (Upasampadā: 24th June, 1979)