We have the idea of Last Universal Common Ancestor now. We are able to track down our tree of life (albeit not really precisely) just by doing statistics on genomes, and the tree seems singular so far. Anything so alien as a different set of amino acids should be immediately obvious (unless it becomes some spectrum rather than separate groups... see next paragraph).
If there is the slightest possibility of eating each other, parasitism or symbiosis, there may be convergent evolution. One may want to eat the other since more food opportunities is better. The other may want to eat the first one for the same reason - you could call it Nash equilibrium, or maybe prisoner's dilemma (they will find out eventually that eating means being eaten, but it will be too late). This may lead to more similarity, eventually muddling the evidence for separate origin. In reality, this can sometimes do wonders like mimicking a foreign DNA sequence rather precisely, without an actual horizontal transfer. A different scenario is one tree of life trying to eat the other, the other TOL trying to remain inedible... This may lead to some interesting evolutionary dynamic, if it can last, which I am not sure. Does not sound like any kind of Nash equilibrium, but, you know, the tree of life is green.
If there is any horizontal gene transfer, I think that implies the level of similarity for convergent evolution to kick in.
OTOH, if they are hopelessly incompatible, there still may be competition for basic resources (carbon, sunlight... unless the other is silicon and geothermal). Competition is enough to drive one side to extinction without any single individual being eaten. Also, not being able to eat each other does not mean not being able to poison each other, for example, so the war can be fought with more direct means. Note that this is not much different from what's happening inside our singular TOL.
In your scenario 3, competition for resources happens even before one of the competitors is around. In an established biosphere, vast majority of resources is locked in some kind of bacterium-bacteriophage cycle (or even in higher forms of life), instead of being available for experiments. A second, delayed abiogenesis may be much more difficult.
Even if incompatible, there still may be high-level convergent evolution, like various harmless snakes mimicking the coral snake. These snakes are not able to fool biologists, though (as long as they are not fossil-only, maybe).
Population size, and probably some evolution rate, affect population stability. A zillion bacteria are more stable than a million dinosaurs. We still do have both bacteria and archaea after all, but no (real) dinosaurs. Both populations starting large and diverse may help; Even if one is clearly superior, the other may find its niche. Again, that happens within out TOL: Diversity persists, no single optimum lifeform dominates. (No, we do not.) If both (all) TOLs have a good start, they all have a decent chance to make it to our biologists, even if it's far from every single species what survives.