My legal middle name is a shortened version of my full birth middle name. e.g Alex legally on my passport, social and so on but Alexander on birth record and marriage certificate. Supposedly City hall by law is required to only use the middle name on my birth certificate and not the name on my social, passport or anything else.
USCIS, like many other government agencies, expect you and everyone use, for that matter, to use their legal name for all purposes. Legal name is the one listed on your birth certificated unless changed as a result of a valid legal process. This does not mean that all people oblige. It does mean, however that all immigrants have to if they are to get positive outcomes from USCIS.
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Niet, don't worry, USCIS does not. What's more, during your interview you'll be able to explain & specify to the CIS officer how is it that you want your name(s) spelled and or listed on your green card.
Kindly be advised that the answer above is only general in nature cannot be construed as legal advice, given that not enough facts are known. It is your responsibility to retain a lawyer to analyze the facts specific to your particular situation in order to give you specific advice. Specific answers will require cognizance of all pertinent facts about your case. Any answers offered on Avvo are of a general nature only, and are not meant to create an attorney-client relationship.
No big deal at all. Use whichever name but also mention the other when asked if you have used any other names.