Correct. Depending on the engine, you either have an idle screw(a.k.a. low speed), or an air bleed screw. They work pretty much the same, though it takes smaller adjustments to get an air bleed type to behave, in my experience. I'm not sure about this particular engine, as I don't currently own one, but they all work pretty much the same.

The large screw on the other side is the high-speed needle, and only affects the mixture at full or high throttle. However, the LOW speed(idle) adjustment will affect both the idle AND the high speed mixture. So when you get your engine to keep running, and preferably with a good idle(by adjusting the low speed 1/8 turn at a time), then you will need to slightly adjust the high speed needle as well. This is what SoonerAce was talking about with the "pinch test".

The "pinch test"... This is how you can tell if your engine is running too rich or too lean at high throttle... With the engine running at full throttle(be careful!), gently pinch the fuel line to the carb briefly, and then let go. If the engine dies completely(stops), then the high speed is adjusted too rich. Turn the (high speed)needle 1/4 turn clockwise, and try again. If it speeds up(high rpm), and/or kind of "stutters", it's too lean, and you should turn the needle 1/4 turn counterclockwise, and give it another try. Ideally, you want the engine to just barely speed up when the fuel line is pinched(that leans it out slightly and temporarily), then return to normal full throttle RPM when released. Listen for the sound while doing this - it's a good indicator of RPM if you dont have a tachometer handy.

As for the Kadet LT-25 - Yes, it's among the best trainers there are! (All the Kadets are good)