I am a home roaster and I've ruined my fair share of coffee beans learning just how to use my own roaster. I use a hot air roaster, the Fresh Roast SR500 and I've done many hours of internet research to try and get a basic understanding of home roasting.
The first basic principle of roasting is that you want the temperature of the beans to be constantly rising. If the beans ever cool during the roast cycle, you've most likely ruined them. It doesn't sound like this is a problem for you.
A second basic principle is that the heat should rise at a rate no faster than the full mass of the beans can absorb. This means that the temperature needs to rise evenly in the whole bean, not just the outer part. If the temperature rises too fast, the outer bean will begin to burn while the inner bean hasn't yet fully dried, so the whole batch is again ruined. This sounds like what is happening to your roast.
Commercial roasters and home roasters using drum roasters are able to roast large batches but at a slower rate than my hot air roaster. This will give a particular flavor profile to a bean that my roaster won't be able to duplicate, primarily because my hot air roaster simply roasts too fast. Nevertheless, I've been able to get very good results with my roaster by learning how to slow it down as much as necessary while still keeping the temperature rising throughout the roast cycle.
I've read in more than one place on roasting forums that first crack (FC) should occur somewhere around 75-80% of the way through the entire roast cycle and should last for as short a time as possible, generally less than 2-3 minutes. My own current roast profile has FC starting around the 6 minute mark and I'm starting my cooling cycle at around 7:40 shortly after FC ends. Not exactly the 75-80% time frame, but those folks are generally using drum roasters in the 15-20 minute range for a complete cycle. They also recommend a roast time no shorter than 10 minutes, but again, that is for a drum roaster. My coffee tastes fantastic with my current profile.
If you get your bean temperatures to rise at a reasonable rate, there is a definite pause between first and second crack, and there is a distinct difference in the sounds of FC vs SC. Since you are using a popcorn popper on a stove burner, I wouldn't be able to give you any better guidance than this, but your 1 cup of beans is relatively small, so I don't think these times are unreasonable. I'm generally roasting 100 grams of raw beans per charge. Your one cup sounds like it's probably I bit more than 100 grams.
I would advise you to go online, search "home coffee roasting forums" and start reading. I would also strongly advise you to start logging each and every batch you do as to times, weight of beans, heat levels, the outside temperature, smells you detect, what color your socks are, etc. Just joking about the socks, of course, but the idea is that you'll begin to see some definite trends and you'll be able to home in on the right method for your setup much more quickly.
In short, definitely slow things down and aim for ending FC around 7:30 or later if possible. End of FC generally equates to a medium roast level, although you'll find lots more information about that online as well. Even with the benefit of the internet and lots of information on the forums, it still took me awhile to get my own setup to produce the results I was looking for so be patient. Good luck!