Does a Mebane address mean you can vote for City Council? No, it does not. You must be within the city limits to vote. If you pay city taxes, you know you are in the city limits. Postal addresses are something very different. I decided to take a look at the actual city limits and thought you may find it interesting. Above is a picture of the city limits, with regions inside city limits in the brighter orange.
You will notice that the city limits have holes and disconnected islands. This is a reflection of how Mebane grows. In order to get water and sewer service, you must be within the city limits. New businesses and housing developments want water and sewer, and so they ask to be added to the city limits (a process called voluntary annexation). Mebane always agrees, and so the city grows, but often in strange ways and shapes.
Mebane includes Mill Creek to the north, does not extend quite as far as Eastern Alamance High School area to the west, and has very little south of the interstate. The area that is south of the interstate is a very recent development and expected to grow more. If you are interested in zooming in and studying this in more detail, I recommend using Alamance County GIS. Google Maps does show the city limits, but it is not very current, so do not rely on that. If you have questions, just ask.