"I am not eating meat, but cheese."
Translation:אני לא אוכל בשר, אלא גבינה.
If you wanted to use אבל , you'd need another verb for it to make any sense. For example: I'm not eating meat, but the cheese is good -- or: I'm not eating meat, but I've got some cheese here... It's like the difference in German between aber (this thing happened, but then that other thing happened) and sondern (not meat but cheese).
note how it is translated as "am eating," which suggests not what the speaker eats in general but what the he is eating right now, this moment. As such it is a contrast. You are correct that since Hebrew doesn't distinguish between the English "I eat..." and "I am eating..." that such a sentence could be used to speak generally about the kinds of things one eats, but you are also correct that that sort of interpretation wouldn't really make sense here. :-)