I'm afraid there's not much way around explaining the German education system at least briefly, if your application is for a job or something where the employer may not be familiar with it. If it's for a purpose such as graduate school, the admissions department is more likely to be familiar with foreign systems, at least at a larger university.
I thought there might be something in the archive underSiehe auch: erster und zweiter Bildungsweg
but as usual, I couldn't find the threads I thought I remembered.
My guess is thatevening school BE = night school AE,
but I'm not sure I would recommend that, particularly if in your case you didn't really go at night. In AE 'night school' is largely outdated, and as you say, it doesn't make clear that you did already have a high school degree, just not one that made you eligible for university admission.
The more modern equivalent in the US is the GED, which I believe stands (or stood; they may have changed it) for 'general equivalency diploma.' It's just a test you pass to show that you have minimal knowledge equivalent to that required for a high school degree.
Can you not give dates for both your other Abschluss and your Abitur, and a short explanation? E.g.,1992 Realschulabschluss (non-university-preparatory high school degree)
1996 Abitur (advanced high school degree required for university admission)