There doesn't seem to be any way of doing a global search on the phone for a certain word (like onpalm or ppc), nor search all the fields in the contacts.

With hundreds of contacts in my phone, this is a must-have feature.

I'm I missing something or am I correct? If so, is there any third party app that does this?

Mobile Search by SEPT-Solutions is exactly doing what you need, a global search for all versions of Symbian