Both of the labels attached to modern numerology assert a lineage, and neither lineage is documented. This is not a marginal quibble; it is the central historical fact about the discipline.
Pythagoras wrote nothing, no text by him was known even in antiquity, and the detailed accounts of his teaching come from Neoplatonist biographers writing seven or eight centuries after his death with a case to make. What can be attributed to him with reasonable confidence is the transmigration of souls and the founding of a community with a rule of life. The Pythagoreans certainly did hold that number was fundamental to reality, and that idea is genuinely theirs. But nothing resembling the digit-reduction of names and birth dates is attributed to Pythagoras or to any Pythagorean by any ancient source. The label is a borrowing of authority rather than a line of descent.
What the modern system actually descends from is documented and recent. L. Dow Balliett published The Philosophy of Numbers in 1908 in the United States, drawing on New Thought and on theosophical ideas about vibration; Florence Campbell and Juno Jordan systematised and popularised it across the following decades. The apparatus of Life Path, Expression and Soul Urge is an American product of the twentieth century, roughly a hundred and twenty years old.
Chaldean numerology, which uses a different letter chart and reserves 9 as sacred, is credited to Cheiro, the stage name of William John Warner, who published it in the 1920s and asserted a Babylonian origin. No documented Babylonian lineage exists for it. It is Cheiro's assertion, and it should be described as such.
None of this makes the practice worthless to those who use it. It does mean that its antiquity is a marketing claim rather than a historical one, and the distinction is worth keeping.