While I can't speak to what is a "legal" name for contract law in a state, the SSA does not consider a suffix such as Junior integral in the name. See this site, https://secure.ssa.gov/poms.nsf/lnx/0110212001 "We do not consider the middle name or suffix part of the legal name. Whether the middle name or suffix is included, omitted or incorrectly shown on evidentiary documents submitted with an SS-5 (Application for a Social Security number Card) does not matter.
NOTE: Both the middle name and suffix, even if not part of the legal name by SSA’s definition, should be used to resolve situations where the identity of the applicant or number holder is in question."
I will say that I have seen contracts that allow a person to use or not use a suffix and there is often a separate part of the contract stating (John Smith is also known as Johnny Smith, John Smith Jr., Johnny Smith Jr., Johnny R Smith... stating variations and have been told by attorney's that variances are common that way.)