Welcome to the forum. :)
It is natural for betta's to flare so that doesn't mean he's stressed. He could also be flaring at you had your face near his tank to look at him. So I wouldn't be too freaked that the little guy is flaring. If he is really small then he is still young. He will become more territorial with age which will lead to more flaring in the future.
Do you mind telling me the size of the tanks? If it's less than a gallon then that could be the reason why the little betta freaked. The little betta only has a little bit of space and a big giant betta could easily take it by force. So, he's putting on a big display to try and scare the big betta away. the big betta doesn't care because the little betta isn't a threat to his territory. I don't keep betta's in anything under a gallon because it tends to make them hyper territorial, ime. You can also find little heaters for 1-3 gallon tanks so you can plug in the heater instead of using a lamp for heat.
If I'm still wrong and you were using the light to make the tank look pretty, then you are using the wrong kind of bulb. There are tank bulbs that screw into lamps for cheap at most pet stores. Maybe changing the regular bulb to a aquarium bulb can make the little guy more comfortable.
I hope this helps. :)