Generally you want the same name in all these places:
- name your references will recognize and remember
- transcripts and degrees
- employer will put on pay cheque
- tax records
- employer will book plane tickets
If all of these are the name you go by, great. If not (eg you had a different last name when you earned that degree or worked at that employer) then in that part of your resume say (as Jane Doe). Introduce yourself, and use on your resume, the name you want to go by unless the large majority of your work-relevant paperwork is in the other name. Deal with tax, paycheque and plane ticket related issues after you are hired.
It sounds like in your case it's the short name that you use everywhere. I wouldn't worry about it being your legal name. I put Kate on applications, contracts, credit cards etc even though my birth certificate has something else. When clients need to book plane tickets I tell them (or their travel agent) what name they need to use for that. As long as you are not trying to defraud you should be fine. A quick sentence in the cover letter that explains why you go by two names might be a good thing if many of your references or resume-related items (eg transcripts) are in the other name.