When playing Team Fortress 2, I noticed that my Strange Flare Gun had more than double the amount of kills my Strange Degreaser had (I started using both of them at about the same time). Even though I'm quite good at aiming the Flare Gun, I found it odd that my primary weapon had a lot less kills than my secondary. This got me thinking; obviously the weapon counts the kills obtained by directly killing using the weapon, but what about kills caused by afterburn damage or kills from reflected projectiles?
To answer your title's question, strange flamethrowers only count kills done by that weapon's fire damage. This is either through direct damage (W + M1) or the afterburn. It does NOT count reflects sadly. Taken straight from the TF2 Wiki's strange page:
As for kill counts, I have a strange degreaser and flare gun too and my flare gun does have a lot more than the degreaser. My guess is in how I play. I tend to ignite, air blast and follow up with a flare gun shot or the axtinguisher. I reflect a lot also (but those kills don't count). If at range I use the flare gun for ignitions and annoying. Since the afterburn damage on the degreaser is less, then you are less likely to get a kill from that than the other flamethowers. Also if you do the air blast and flare gun, the flare gun's afterburn will override the degreaser's so if you get the kill after that, it's for the flare gun.
I would so love a separate counter on my degreaser for reflect kills, like the sentry kill count on the engy's wrench.