A short answer to your question, “no”, there is no 'definitive' Pipe arms.
There is no such thing as a Coat of Arms for a family. Arms could be granted to an individual, which may possibly be hereditary in some instances via a direct line (eldest heir to eldest heir etc).
In the past several arms were granted to various Pipes, all of them originating from the Staffordshire line. There is no evidence that any Pipe from Suffolk had any right to arms, and most of the lines of succession from Stafford appear to be extinct.
The FIRST one that you refer to is one that is touted on the Internet by various irresponsible companies who do absolutely no research to back up their claims. If they did their research they would find that this one was granted to Sir Thomas de la Pipe (Staffordshire) by Edward II around 1320, described as “Azure, crusily 3,3, and Fess Or” in the Parliamentary Roll. This line has no descendants as far as is known, so the arms are extinct.
The arms that are most usually associated with the name of Pipe are others that were granted in Staffordshire and are totally different from those that you describe. The line also became extinct when the only heiress passed her inheritance to the Vernon family, the arms going with it. These arms c 1380 (Azure, crusily, two pipes pileways Or) can be seen at various places in Haddon Hall, Derbyshire, and also in Tong Church, Shropshire, on the tomb of Sir Richard Vernon (d 1467) and Margaret Swynfen his wife, heiress of Sir Robert Pype.
Sir Richard Pipe, Lord Mayor of London, also a descendant of the Staffs Pipes, was granted arms around 1587, but this was his grant, not the Pipe name as a whole.
Another line from Staffs, Pipe-Wolferstan has the right to arms since Jun 8th1776, which also contains Pipe, but that is for one specific line.
The second one that you describe is from “Burke’s General Armory 1848”. The description is:
“Sable, on a bend Or, between two nag’s heads erased Ar, three fleur-de-lis of the first.”
It is attributed to Pipe, Cottenham, Cambridgeshire. This one gave me the run-around for some time, until I finally discovered that it had been attributed incorrectly to Pipe, and was in fact the arms of Pepys, as in Samuel Pepys, who was from Cottenham.