This is how I understand it as well.
A tempo run is a pretty vague term, tempo just means "pace"...there are a lot of different paces.
To me its just a high end aerobic run, faster than base miles, but slower than threshold (1 hour race pace).
The goal of running them is to teach yourself to run faster for longer distances. The benefits are the same as a general aerobic run, but you get more benefit for the time spent. Done correctly, you should running as fast as possible while remaining completely aerobic. Any faster becomes a threshold run, which has a different purpose (increasing the point where your muscles produce lactic acid).
You end up splitting hairs though. Its not a hard cutoff between aerobic/tempo or tempo/threshold. Its a continuum. This is why its better to just do them (most training, if we are honest) by feel. Not too easy, not too hard.