Well, I got it working myself. I'm not sure what was wrong the first time as I started from scratch with a new HTML file. I hope it will help someone:
<!DOCTYPE html>
<html xmlns:fb="http://www.facebook.com/2008/fbml">
<head>
</head>
<body>
<a href="#" onClick="postToFacebook()">Post to Facebook</a>
<script>
function postToFacebook() {
var body = 'Reading Connect JS documentation';
FB.api('/me/feed', 'post', { body: body, message: 'My message is ...' }, function(response) {
if (!response || response.error) {
alert('Error occured');
} else {
alert('Post ID: ' + response);
}
});
}
</script>
<div id="fb-root"></div>
<script>
window.fbAsyncInit = function() {
FB.init({
appId : 'YOUR APP ID GOES HERE',
status : true, // check login status
cookie : true, // enable cookies to allow the server to access the session
xfbml : true // parse XFBML
});
};
(function() {
var e = document.createElement('script');
e.src = document.location.protocol + '//connect.facebook.net/en_US/all.js';
e.async = true;
document.getElementById('fb-root').appendChild(e);
}());
</script>
</body>
</html>