Dave’s “big list” entry for “dollar” seems to split the difference between a Dutch/German ancestry, referring to “daler” as a Dutch / Low German form of “taler” (itself a clipping of “Joachimstaler").
I am inferring from the big list entry that the American colonists borrowed the term “dollar” from British usage (and applied the term to the peso) rather than having independently adopted the term “dollar” from the Dutch settlers.
I haven’t, of course, done any independent research on this, and may also be reading too much into the “big list” entry.
In any event, “dollar” strikes me as an interesting illustration of the difficulty of defining what constitutes a single word vs. multiple words. Is the British “dollar” (informal name for a crown) a different sense of the same word as the American “dollar”, or are they entirely different words which share some etymological features? How about “dollar” (for crown) vs. “dollar” (for undefined amount of money) vs. “dollar” (for a Joachimstaler) vs. “dollar” (for a coin that was modeled after the Joachimstaler)?