I got the same problem and I think I found the problem. As all the content is loaded by the AJAX request also the <script src="file"></script> parts are loaded via AJAX But the Browser doesn't load the files in the script tags and therefore he can't access the functions. if you include all the necessary files and settings before you get the content with AJAX it works.
So you have to include the following script tags within your page which loads the AJAX content:
<script language="JavaScript" src="CuteEditor_Files/Scripts/Constant.js"></script>
<script language="JavaScript" src="CuteEditor_Files/license/CuteEditorPHPLic.js"></script>
<script language="JavaScript" src="CuteEditor_Files/Scripts/Gecko_Loader/Loader.js"></script>
<link href="CuteEditor_Files/Themes/Office2003/style.php?EditorID=CE_description_ID" type="text/css" rel="stylesheet">
<script language="JavaScript">var CE_Editor1_IDSettingClass_Strings={"......};</script>
But the problem with this solutions is the broser defined Loader.js so you have to check yourself which browser is used by the user of your page.
So it would be nice to have a DrawOnlyScriptTags() like function to preload the javascript files and then initialize the editor from the AJAX request.
As I only have a download link to CuteEditor_PHP_**********.zip I wasn't able to test the 6.1 version. I just purchased the editor two weeks ago, so if there is a 6.1 version I would be happy to get able to use it.
@Adam: How can I get AJAX working with the 6.1 version?