On Ookla: This approach does not seem very systematic. The whole "bandwidth measurements using HTTP from flash, within a browser" is suspect to begin with. They do at least (implicitly) acknowledge this. Strange, though: why not report the whole distribution, and let people draw their own conclusions. I agree, very odd.
On the FCC test: the validity of a warmup period in some sense depends on what you're trying to measure/report. If TCP is in slow-start at the beginning of the measurement, an "warmup" is simply ensuring that the measurement reflects steady-state TCP throughput, I could sort of understand the approach. In that sense, you're getting a more accurate reflection of link speed, and it is often the case that it may also reflect real user experience, too (e.g., many browsers try to get out---or stay out---of TCP slow start since the connections are so short). Slow start will make the connection seem slower than it is in steady state, especially if the test is short. Another way around what they're doing could be to simply open up a bunch of TCP connections in parallel---that would fill the pipe faster and perhaps give a more sensible reading, even without "warmup".