I did some more research and I can confirm that all button elements have to be within the <form> tag to work properly.
It also seems to be impossible using two {NAVIGATOR} placeholders, because the second one never shows up.
Instead, you can have your own "next" (and probably also "previous") button which you can hard-code at the template files (within the <form> element):
<button class="submit" type="submit" accesskey="n" onclick="javascript:document.limesurvey.move.value = 'movenext';" value="Next" name="move2" id="movenextbtn">Next</button>
You should additionally add a line of CSS to hide this custom "next" button on the last page because otherwise, there will be the standard "submit" button plus another button names "next".
I have created a copy of the test survey ("Test survey Marcel (ID 518962)") and assigned a copy of the template for testing ("praclabnew_marcel").