I have never experimented using 3 x AAA NiMH battery substitute for 18650 battery, however my hunch is that the answer will be yes depending on various factor as described below.

The maxium discharge rate for the 18650 or the 26650 is much higher than those for the AAA NiMH battery assuming you will be using better know brand 18650 cell like Panasonic, Redilast. Some of the cheaper brand like those from Ebay will only perform limited X amount of current discharge before the internal resistant affect the performance. Even with that, those cheaper brand 18650 still can produce fair amount of current discharge and most likley be more than those of AAA NiMH. This still may not mean you will get better performance from your particular flashlight. It will depend on what the driver that was used for your flashlight. If it is one of the more typical high lumens flashlight where they are driving the LED at a high current, then I can say for sure that the 18650 will be the way to go. It should provide longer runtime and higher current for the driver to drive the LED. If the flashlight has a low current driver and the flashlight spec is one of the lower lumen flashlight, then I will have to say that the 18650 will provide much longer runtime but it will not provide anymore lumen since that is already spec out by the driver. The listing you provide was 1800 lumens which pretty much tells me that it most likely have one of those drivers that drive the LED at 2.5 to 3 amps. In this case, you will benfit from 18650 or 26650. What is your current runtime when using 3 AAA NiMH and does it dim after a short runtime?

The difference between 18650 and 26650 is not much. There are now 3400 mah 18650 battery out in the market while most 26650 are around 4000 mah. There are listed 26650 with much higher rating but I would not trust those listing. If this was to be used as a helmet flashlight, stay with the 18650 mostly because of the weight difference. 26650 is around 100gram and the 18650 with the adapter is about 65 gram. That 25 gram can noticable to some folks when these things are on top of their head. For use as a handle bar light, personally I like to use the 26650 battery instead. For one thing, you can get 26650 4000 mah battery cheaper than the higher quality 3600 mah 18650. One thing worth noting is that the performance from a 4000 mah 26650 will be overall better than from a 3400 mah battery beside the obvious higher rating. The 4000 MAh 26650 can yeild a higher discharge rate than the 3600 mah 18650 and that will help provide less of a voltage sagging which can dim the light.