"Different first page" should do what you want. However, if there is a section break in the document, you may actually have two first page headers (one per section), and you will then have to unlink the headers.
Note that Word can have up to three different headers (and footers) per section--first page, even page, odd page. This is the case if you have selected both "Different first page" and "Different odd and even" in Page Setup (or on the Header & Footer Tools Design tab of the ribbon). See
http://sbarnhill.mvps.org/WordFAQs/HeaderFooter.htm.
Specific information on how to create a letterhead can be found in the article at
http://sbarnhill.mvps.org/WordFAQs/Letterhead.htm.