I don't seem to remember College transcripts showing your incoming admission status (i.e. International), but I may be incorrect it's been a few since I left my job evaluating transcripts. Transcripts show grades, classes, standing as in grade level, enrollment status, and previous education. But let's say it does show app status of international, from that time forward all other schooling would also enroll him as international until he became a US Citizen at which point there would be a fairly lengthy paper trail both within the government and within school.
Since Obama received Federal Student Aid it would have been impossible for him to be an international student. Now what often happens is a student will make a mistake and file an international admission application, usually the admission team catches this and it's fixed pretty quickly. In the case of Obama with his educational background it could have been missed and then fixed when admissions and the international students office requested visa applications and financial solvency or financial aid would pick it up when applying for aid. If financial aid picks it up, in order for them to change anything and disperse funding the student has to provide proof of citizenship. On occasion an admissions team will see someone graduated from a foreign high school (a child of a us diplomat) and assume the student is international and request a new app or proof of citizen. I've also seen financial aid incorrectly label a student international based on their address at time of filing and again this usually requires proof of citizenship.