It's called WordprocessingML (aka. WordXML, WordML)...
It could be called XForms...
The Web was suppose to be platform-independent electronic documents. In other words, if you truly want platform-independence, then I agree with you and your forms should be in HTML. Yet, HTML forms are really not a good development platform. That is why Adobe, Microsoft, and others provide "form" solutions. XForms is an attempt to make developing and using HTML forms more flexible, overcome its limitations, and provide a platform-independent object model for completing HTML forms. You might want to look at XForms at http://www.w3.org/MarkUp/Forms/.
But, I wouldn't call it PDF
In my opinion, working with PDF files is difficult. I have not looked at the file format specification, but I heard it is not trivial. Moreover, you need a custom editor and you are locked into one vendor, which is Adobe. (Yet, there are other open-source and vendors who support the file format.) Adobe is not know for creating programs that are easy to use.
My Suggestion
If you are already using Word, then moving to WordML should be fairly easy. You can easily convert your existing Word documents into WordML by simply saving them as XML from the Save Dialog; therefore, you can automate this process through code. In addition, I believe WordML supports form templates (the actual form) and data documents (the actual data for a form).