As mentioned in previous threads on this topic, Tamsui is a historical name.
The Spanish had multiple renderings of that name as soon as they ventured out of San Salvador (Spanish fort on Heping island) sometime after 1626.
After 1630, Dominican Friar Jacinto Esquivel wrote two dictionary for Tamsui area aboriginals called "Vocabularino de la lengua de los Indios Tanchui en la Isla Hermosa" and "Doctrina cristiana en la lengua de los Indios Tanchui en la Isla Hermosa."
Other renderings include the Spanish's Tanchui, Tamsui, Dutch's various renderings, Tamsuy, Tampsui, and Tamsay.
The person that really cemented the Tamsui spelling was George Leslie Mackay, who wrote extensively about the town in his book "From Far Formosa"