Basically I do not know and can’t claim to know what name he used at Occidental. The poem you quoted corroborates that he was indeed using the Obama name that year. Is the source of that poem and name contemperaneous - if so, then that year he was using Obama. What name did he used when he applied? The same name? Do you know?
And even more important than the name, is the nationality.
It’s possible but doubtful that Stanley Anne changed Barry’s Indonesian name from Soetoro to Obama —again, possible but doubtful. When do you think this would have happened, 1970? How close was Stanley Anne to the Indonesian magistrates at that time?