As a term of art, "Notice of Intended Prosecution" does not appear anywhere in s. 1 (or 2 for that matter) RTOA 1988 or s. 172 RTA 1988.
We have seen numerous cases where conflating a written warning and a requirement for information under s. 172 RTA 1988 has caused confusion or issues - mostly the potential for OPs to wishfully think that a late NIP (particularly when they are not the RK or haven't updated the V5C) means they can simply bury their head in the sand and ignore the incorporated s. 172 requirement, but in the context of this thread, this is pure pedantry. If you wish to engage in such pedantry, may I suggest that the next time a motoring case comes before you you ask the prosecution to show you where in Schedule 2 of the Road Traffic Offenders Act 1988 it prohibits whatever it is the criminal scum before you is accused of doing?
In the alternative, a NIP almost invariably incorporates an s. 172 requirement. Whilst the title of the combined document, which as previously mentioned does not exist as a term of art in the legislation, is descriptive of the requirement under s. 1 RTOA 1988 which the first such notice in the chain frequently purports to satisfy (where such a requirement applies), the combined notice is served in some cases to satisfy the requirements of s. 1 RTOA 1988, and in almost all cases to engage the obligation under s. 172(7) RTA 1988 - so the notice was sent "under" s. 172 to the extent that the word "under" is meaningful. S. 172 does not in and of itself "require" a notice to be sent out, unless the police wish to engage the obligation under s. 172(7), in much the same way that s. 1 does not require a NIP (or other form of written warning without that term of art as a title) to be served, unless the police wish to prosecute the driver and have not verbally warned him at the time of the commission of the offence. So, if you wish to engage in pedantry, make sure you get it right - but mostly the Schedule 2 thing.