I am not sure if you are talking about the fluff or the visuals or the mechanics.

But I still don’t understand how this accounts for the concept of a single player bullet on the screen.

Your single bullet at a time still is a threat to the enemy for a short duration. Repeatedly.

So whatever different weapon you would implement, it should be cooldown based and have a finite duration.

So if you want an effect in front of the ship, it would appear, stay a short amount of time, and dissappear. It's advantage would be spread/area coverage, it's disadvantage range. Basically it is a melee attack.

The current bullet you fire is traveling not instantly. It's advantage is range, it's disadvantage is flight time and thus inaccuracy. The cooldown is implicit, since you cannot have more than one bullet on screen. Which would make shots near the edge of the screen be better, since you could fire sooner - or maybe there is a cooldown independant of a bullet on screen, I do not know. Basically it is a slow ranged attack.

A fast ranged attack would be a laser shot. It's advantage is speed, it's disadvantage is timing. While the bullet flys slow enough to dodge, it also flys slow enough that you have more opportunities to hit the opponent that crosses the flight path of the bullet.

From mechanics and fluff, I prefer the booster burn. Go to close range, turn or pass, or lure enemies to fly behind you, and burn your opponent by ramping up the thrusters. This might also be used to dodge. And it would be closest to what a flame thrower would feel, plus paying tribute to the mechanics of a free fall space vessel.

And that is what I meant with energy flow. If you do have particles in a stream exhausting a space vessel, that would influence it's flight path. It would matter little, if the gas is your air or is burning. I believe that is literally rocket science. The burning exploding would just propel the gas faster and give more thrust. 

If we go away from "flames", plasma comes to mind. It is hotter than fire and does not burn up oxygen. It basically is just hot matter. Very hot. Solid, liquid, gas, plasma. And since you can influence it electrically/magnetically, that would be a way to handle it. A solar prominence is plasma flowing along magnetic lines. There are also toroidal structures such a plasma can have.

So, hurling plasma at your enemies might be made to look similar to a flame thrower's mechanics.

The V thing is a different concept, I think it would not work in space. But a plasma filament might expand in a similar manner, if guided by magnetic fields, see solar prominences again. It is V shaped electrodes. Or rather \ / and between those you apply a strong electric force. There will be an arc travelling from bottom to top between the wires.

Mechanical and visual, you might want to either have a  \ / area or a |  | area of threat. The area would be a death zone for a short time with a recharged period. There could be an ellipse traveling from bottom to top. Or mirrored effect of your thrusters. Or any other effect like desribed above from those old games. Range, recharge period and how long it would be a threat zone, would be the balancing screws to make it comparable to a bullet as a weapon.