Would running a heater for 10 seconds on then 10 seconds off be the same as alternating on and off every 1 second for 20 seconds?
Your premise is largely correct but there are a few things to take into consideration:


Temperature coefficient of resistance causes the heater resistance to increase with temperature. The primary observable effect of this is that on switch-on the current is initially somewhat higher but decreases as the heater warms up. This is most famously observed in the incandescent bulb whose turn-on current can be five to ten times higher than its steady-state current. The effect on your heater would be to increase the average power slightly.
Thermal response time is the rate at which the load temperature reacts to the power turning on and off. In most industrial applications a steady temperature is required / preferred but on/off proportional power control is used (as in your question) for simplicity and reliability. The solution is to switch with a cycle time that is short relative to the response time of the system. e.g. If the system is at operating temperature and turning on the power results in a temperature rise of 10°C / minute and we want to hold the temperature to within 1°C then I would expect that running at a 2 or 3 s cycle would manage that.


Mathematically, you are putting energy into the system and this can be calculated by \$ E = \int P dt \$. i.e., the area under the power-curve.