My ancient family name was Pootman, or Potman, which seems to come from Aalburg, North Brabant, Netherlands, but before that perhaps the Lower Rhine area of Germany, where the name Puettman is very common.
As far as I know the name Puett comes from Putt with an umlaut over the "u" and means a pit or mine.Putt with umlaut in German is a pit or mine.It is related to the French Latin word "puteus" meaning a well.The closes English word for it is "pit"
Since a pit often fills with water forming a pool, the name Putt is sometimes said to mean pool.German "pfutz" means puddle or pool.Actually, the best Germany word for pool is "pfuhl" or a mud hole.[All "u"'s with umlauts.]
As to Pootman, the name is seen mostly in Netherlands and Belgium as meaning Portman or Gateman, but in Lower Germany the name Pootman seems to mean a person who lives near land that that is inundated in the spring with water or a "pool forming area'.Then Pootman in late case means Poolman and may be connected with the earlier name Puettmann.
"Put" in Dutch means well or pit, and you seen place names such as Putten in Holland meaning the place of the well.
In Germany, the word for a "pot" or jar is usually "topf", and a potter is a "topfer".
My best guess is that Puett [Putt with an umlaut] means in Northwest Germany pit or well and perhaps pool.
It is commonly said in English areas that the name Putman means Pitman, which is an extension of Puttmann or Puettman.All these names are related to Latin Puteus . . . a well or pit.