Middle distance runners run typically everything from the 400 to the mile, but most middies prefer to run the 600, 800, or 1000 meter races. These races are longer than the typical sprints, the 50, 100, and 200, but shorter than the typical distance races, the 1600 (aka mile), 2 mile, and 5000 meters. Within the context of this song, the term middle distance runner refers to his ability to maintain a romantic relationship. It is not a fling, or a sprint, but it isn't the slower paced, long lasting distance race, which would compare to a long term relationship.
Middle distance runners are much faster than distance runners, but lack stamina. In the metaphor of the song, this means the relationship is more intense and passionate, but shorter. The lyrics back this idea with the lines:
"So won't you run to me tonight
Tonight let's not talk about next summer
Cause I'll only ever be
A middle distance runner"
and
"But you'll have to run to me tonight
Tonight I will love you forever
But I'll only ever be
A middle distance runner"
With the second section however, the lyrics show the contrast between a sprint and a middle distance race, saying:
"So won't you run to me tonight
Do not pretend that we're just lovers
But I'll only ever be
A middle distance runner"
It is clear that the protagonist wants more than a fling, but cant promise a life of attachment.