Okay, here's what I was taught by a master mechanic with over fifty years of experience:
There are two 'proven' ways to break in a new engine or to seat new rings.
The first, and most common method:
Run the motor in short cycles (30min-1hr) at a time, never going above half throttle. During this break-in period, you will want to run a 30-50% higher mix of oil:fuel. (IE, if it calls for 20:1, you're best bet is between 14:1 and 16:1)
This method has quite a few benefits. The main one is that with a smooth break-in, you promote the longest lifespan possible out of all of the components. You also aren't ripping excessive amounts of material from the rings. For a daily operated motor that won't be 'raced' and just used in a normal manner, this is the absolutely best method, tested over time.
The second method:
Start the motor, allowing it to idle for a few minutes to warm it up. Then, with a load on it (NEVER break in a motor without a load against it), dial it up to 50% throttle for two minutes. Allow it to drop to idle and rest for two minutes. Then dial to 75% throttle for two minutes, and allow to idle for two. Dial to WOT for two minutes, and idle for two minutes. Run at 75% throttle for five minutes, idle for two. Run at 50% idle for five minutes, idle for two. Allow to cool completely before running again. DO NOT force cool the engine with water or compressed gas. Natural cooling only (if in winter, have a space heater nearby so that the motor doesn't cool too fast. I would also use it to warm the incoming air if it's below 40F, so that the oil isn't condensing and creating 'pockets' in the mixture.)
During this process, you will want to run a oil:fuel mix with 50-75% more oil. Also, before initial run-in, remove the spark plug and put four drops of oil in a NSEW (ninety-degree spacing basically) around the rim of the piston. Allow a few minutes for the oil to soak in and spread. If you do not do this, you will end up dry-running the engine (little or no oil being transmitted to needed areas) and burn off the rings.
This is what is commonly referred to as a 'racing' break-in, or a hot break-in. The benefits to this is that the motor has about a 20% chance to gain in the top end, and can handle heavy loading (fast cranking of the throttle) or being operated at WOT for longer periods. One of the down sides though is that it will wear more material from the rings. This means they will need to be replaced faster than rings set in with the more conventional method.
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When running a motor in, you are using the compression generated to 'push' the rings into position, the hashing to smooth the ring surfaces, and the heat from the motor to cook them into permanent shape. All of these things must be factored in if you wish to set the rings properly and have a motor that will last you.
Utilizing the first method, I have run-in two cycle engines that have seen tens of thousands of hours of operation with only basic maintenance required. Spark plugs, a coil or two, things like that. The second has also produced great-running engines, though generally they have needed tear-down faster.
It's all about what you want, and what you feel comfortable with doing.
Also, talk to a local small-engine shop mechanic about your choices. They will always have little tid-bits and tricks to help you get the most from your engine as you break it in to get exactly what you want from it.
And remember, no two engines are perfectly alike. What works for one motor might not be the best option for another. Some engines will run best with a slow run-in, others with a hot one.
