I'll add my thanks to that, Uncle Cheese. It worked beautifully once I figured a few things out that are probably obvious to most people but not to a relative beginner like me.
In case anyone else gets caught out, when you copy/paste from the source code the action in the first line will have a specific URL (eg action="/page/subpage/SearchForm") and it may not give you the results you want (since I had copied from a dev page it was asking for a login before it would show results). Use {$Link} to get the current URL path.
<form id="SearchForm_SearchForm" action="{$Link}SearchForm" method="get" enctype="application/x-www-form-urlencoded">
Add your placeholder="Search ..." attribute to the text input and the rest of the form code is OK as is.
Then to add your customised search from to each template, replace $SearchForm with <% include MySearchForm %> (Yes, obvious I know; at least this one only took me a few seconds to work out!)