And you are partially correct.
I have posted this in many locations to inform people of what can happen to your battery due to the signal.
It is not just a cut and dry low signal more power thus higher battery drain. nope..
The other two factors are radio switching. Yes this is a bad thing and it happens a lot in the US due to LTE signals being weaker as they are higher in the spectrum. So what happens that can cause a battery drain in a fluctuating signal location.... lots.
First when ever the LTE signal drops to a lower signal the phone will wake up completely(not the screen) to do a dynamic check of the 3G signal then compare and switch. This process can take 5-20 seconds and usually both radios will turn on and go full on transmit power to poll all towers in the area. Now imagine if this happens a lot because you are in a crappy "fringe" signal area of 4G. You guessed it tons of high powered battery draining CPU full on system wake lock till it can decide what to do.
In the US this happens a lot, well, because we have LTE but the coverage is not that great. Plus the 4G radio will wake and poll every minute to see if there is a signal, but usually this wake will be a listen only mode with no transmitting.
The second thing that will kill your battery is the bandwidth. If an app has to wake due to a push notification then has to hit the server to retrieve the info that takes time. That time spent retrieving whatever it was that it received a request about can be big or small. regardless of that it still needs to pull data down. This can be bad if the tower your getting your data from is crowded or old/out dated and slow. The longer the app has to run the longer it drains the battery and if the bandwidth sucks even though the signal is great then your going to have battery drain.
Now why is the second one important. Well you have a smartphone. It polls and runs apps and retrieves data in the background all the time. If it has a hard time doing this then it has to stay awake longer to get the job done and thus battery drain.
What can you do about this... well nothing short of finding a carrier that has better signal/bandwidth/coverage in your area. Keep WiFi on all the time so it can latch onto any open WiFi it can find. Or turn off your data, which of coarse kills the point of having a nice smartphone that is always keeping you up to date.
Remember that 3G is still on even when LTE/WiFi is on, but if 3G in your area sucks for speed(bandwidth) then you'll likely have greater battery drain do to wake locks and therefore leaving LTE/WiFi on would be the lesser of two evils.