Fair enough, if you're doing it for the sake of knowledge, more power to you. In the interest of science, here are some other things to consider:
Thermostats of at least one major manufacturer I am familiar with (wink, wink) use half a byte to store the partial degree. So, your thermostat may display 63 degrees but the actual ambient air temperature is somewhere between 62 8/16 and 63 7/16 degrees. So for your experiment, use a more accurate thermometer and ignore what the thermostat says. If your temperature differential of your thermostat is +/- one degree, a thermostat set at 63 will issue a call for heat at 62 7/16 degrees (or lower) and will end the call for heat once it observes a temperature of 63 7/16. Again, all it can do is suggest to the furnace that it turn the heat on or off, so the furnace logic board will also determine if the furnace starts up instantly and how much longer it runs after the call for heat is terminated based on whatever logic the furnace manufacturer coded into it.
As I mentioned before, household thermostats also have some delays built in as "sanity checks" so sudden temorary gusts of extreme temperature (open the front door in winter, open the oven door, running a hot vacuum under the thermostat, etc.) do not affect the display temperature or operation. You can get a sense of the delay built into your thermostat by getting a hair dryer and blowing hot air on your thermostat. If the hairdryer is blowing 130 degree air, it will take several minutes for the display temperature to climb, and the display will most likely only climb a few degrees at a time because thermostats assume that a normal house only adds or drops a degree every 10 minutes or so. Since you don't get to view the partial degrees the thermostat is using to decide what to do or know what the sample rates and tolerances are, it will be difficult to time your manual adjustments with any sort of precision.
And lastly, electronic thermostats have a delay to deal with with the user input. If you have a setpoint at 63 and you hit the temperature up arrow 10 times to get it to 73, what is actually happening is something like: <user interrupt timer start><setpoint +1><reset interrupt timer><setpoint +1><reset interrupt timer><setpoint +1>... <setpoint +1><wait for user interrupt timer to expire><return control to tstat logic board>. The user interrupt depends on model but is usually somewhere between 5 and 15 seconds. So your point of having to start your measurements based on the audible click of the call for heat is correct, but you will also need to account for the extra delay when turning the heat setpoint back down.
You might also not be aware that gas furnaces do not always light immediately ON PURPOSE. In order to ensure that the ignitor is using the lowest possible current and the safety systems are working, the furnace logic board every X ignitions or so (frequency depends on manufacturer) will "reset" the ignitor to the lowest possible current, then attempt a series of ignitions, gradually increasing the current to the minimum necessary until the fire starts. This saves the wear and tear on the system in the long run. So you actually do need to observe the gas fire while listening for the click from the call to heat to determine when the true heating cycle starts to get a proper timing.