Is there a convention for referring to a scholar's academic status? I'm currently preparing an online tool for scholars, and part of that work involves creating a database with different categories for students at different stages of their college/university education (i.e. freshmen, second-semester juniors, first-year graduate students, etc). I'm struggling to find a standard classification system to use that would allow distinction between all of the stages in an undergraduate and graduate education. I could just use "first-semester senior" and the like, but if there's an existing standard - particularly one that even allows for students that don't use semester classifications - I don't want to complicate things with my own classification scheme.   

Is there a convention, especially an international convention, that distinguishes between all or most of these stages in educational progress?
There is no international standard. You will at best find national standards. The problem is that an undergraduate degree can range from 3 years (there may even be shorter ones but I have not heard any examples) to 4 or 5. In some countries a masters degree is a graduate degree in some it is an undergraduate degree. moving to graduate school, Phds range from fixed three years through fixed four years to "individual length. A masters is not necessary to do to move from an true undergraduate studies to PhD, which means it will be an optional step in an education for some.

This means you will have problems defining a solid difference between undergraduate and graduate with the masters belonging to both. You can of course call students by the year they are at at each level but it would probably be beneficial to then know if it is m'th year out of n years to be possible to evaluate. The terminology for undergraduate students at different levels common in, for example the US, which you mentioned in your question is certainly not used internationally.

Based on this "mess" you should probably try to find a system that is as simple as possible and, for example, in the case of the masters simply define it for the site as undergraduate or graduate. Referencing to the year into an education will probably be understood by everyone. There are attempts, for example, within the European Union to synch levels and educational systems but so far there is much left to do to reach such goals if it ever will happen.