What is apex:commandButton with apex:actionSupport?

When using it actually submits the page. You could see the HTML button actually type is submit.

Refer documentation

A button that is rendered as an HTML input element with the type attribute set to submit, reset, or image, depending on the tag's specified values. The button executes an action defined by a controller, and then either refreshes the current page, or navigates to a different page based on the PageReference variable that is returned by the action.

That's the reason, it is submitting the page and re-initiating the counter
Workable code will be (without actionSupport):
Here with the use of action you can submit the page and increment the counter and use the rerender attribute to refresh the counter.
       
Otherwise putting action="{!incrementCounter}" in commandbutton will keep the incrementCounter but it makes no sense to use actionSupport with the commandButton in this scenario.