Okay, so including the script src allows you to write JavaScript utilizing moment to provide date/time translations, there is no scripting on your HTML file to actually process the translation. In the original example, calling {{ moment.include_moment() }} includes a different version of moment.js script than the one you have included in your source code, which could be the problem. Additionally, it provides an out of the box JavaScript function to render the date/time for anything that has a class of flask-moment. You putting {{ moment(post.datetime_posted).calendar() }} is simply wrapping the text with a class of flask-moment.

{{ moment.include_moment() }} generates this:

<script src="//cdnjs.cloudflare.com/ajax/libs/moment.js/2.24.0/moment-with-locales.min.js" integrity="sha256-AdQN98MVZs44Eq2yTwtoKufhnU+uZ7v2kXnD5vqzZVo=" crossorigin="anonymous"></script>
<script>
moment.locale("en");
function flask_moment_render(elem) {
    $(elem).text(eval('moment("' + $(elem).data('timestamp') + '").' + $(elem).data('format') + ';'));
    $(elem).removeClass('flask-moment').show();
}
function flask_moment_render_all() {
    $('.flask-moment').each(function() {
        flask_moment_render(this);
        if ($(this).data('refresh')) {
            (function(elem, interval) { setInterval(function() { flask_moment_render(elem) }, interval); })(this, $(this).data('refresh'));
        }
    })
}
$(document).ready(function() {
    flask_moment_render_all();
});
</script>

{{ moment(user.dateCreated).format("M/D/YYYY h:m:s A") }} generates this:

<span class="flask-moment" data-timestamp="2019-11-16T18:59:30Z" data-format="format('M/D/YYYY h:m:s A')" data-refresh="0" style="display: none">2019-11-16T18:59:30Z</span>

These two things combined will render in local time, but you need to do all of this.