How should I add sign up forms into 3rd party sites?
Non-programming approach:
Use HTML Snippets for the profile. If it is not working according to documentation in book it might be that a) the documentation needs to be updated, or b) there is a bug in this functionality, or c) there is a bug in the custom page that includes the snippet. I would expect that c), b) and then a) are the likely order of solving this.


NB: when using HTML Snippets, you should ensure that permissions to create all the objects needed for the form (eg custom fields, contacts, etc) are available to an anonymous user.


Programming approach:
a) If you are doing this to allow one other site to callback into CiviCRM, then I would have it call CiviCRM's API via REST - http://wiki.civicrm.org/confluence/display/CRMDOC/Using+the+API#UsingtheAPI-REST.
b) If you are expecting to reuse the form in multiple other sites, then I would create an iFrame via your CMS and include the iFrame from the other sites. The iFrame itself would still need to use the external services protocol at the above link rather than AJAX since in general it will not have logged in before posting.