I go with processing and developing film as effectively interchangeable, but tend to subconsciously go towards 'processing' if talking about more complicated or extended steps, and developing if nothing overly interesting is going on. However I'm horribly inconsistent with it, and would like to apologize to any non-native English speakers (and probably any non film enthusiasts) who ever get to sit down to talk photography with me.
If I'm going to develop, review, make notes, scan, catalog, keyword and cross reference, then I tend to think "Processing", and the act extends well beyond just the chemistry stage. Processing may also mean dealing with stuff that has already been developed and sat in the in-box for the last week or two while I dealt with other things.
Language is fun and wonderfully inconsistent.