I'll assume that you do a good job of using various clues (the time you see the light, your location when you see it, the direction of the light, and some estimate of the distance to the star) and correctly work out more or less where each explosion took place in spacetime.
In this case, no matter where you are, no matter your speed, and no matter anything else about you, you will derive the same spacetime locations for the explosions, because the locations are an objective property of the external world and we're assuming that you measured them correctly.
There are a lot of different ways you could write down the locations. Those ways are called coordinate systems. Some coordinate systems have a coordinate called "t" in them, and depending on the coordinate system, the t coordinates of the two explosions might be the same or might be different. This isn't a property of the explosions, but of the arbitrary choice of coordinates.
The choice of coordinates is really up to you. In introductions to special relativity it's common to assume that everybody picks an "egocentric" coordinate system (a term I just made up for coordinates in which they're at rest at the origin). If everyone does that, then people moving at different speeds are likely to disagree about the equality of the t coordinates of various things. But (if they're good scientists) they won't disagree about the objective locations of those things, because they'll understand that their choice of coordinates doesn't influence the actual locations. And also (if they're good scientists) they'll understand that they don't need to choose egocentric coordinates, and especially if they're collaborating with someone else they'd be better off agreeing on a common coordinate system to communicate their results.
As I said in my other recent answer, in general relativity the choice of useful coordinate systems tends to be more restricted because most spacetimes have fewer symmetries than the Minkowski spacetime of special relativity. You can still use any coordinates you want, but most will be inconvenient because the metric will have an unnecessarily complicated form. In particular, it tends to be inconvenient to use egocentric coordinates.
When you're talking about the universe on a large scale, only one time/"t" coordinate, called cosmological time, is convenient, because it's the only one that respects the large-scale spacetime symmetries of the universe we live in. When you see a "time since the big bang" in articles about cosmology or astronomy, it's cosmological time.
When you work out the coordinates of the two stars, you'll probably end up with the same t coordinates as someone else working independently on the same problem, because you both will pick the most convenient t coordinate, and that's cosmological time. It doesn't matter where you are or how fast you're moving, since it's dictated by the objective "shape of the universe" which everyone agrees on in principle, if they have accurate enough equipment to measure it well. You could pick a different coordinate system and disagree with the other experimenter, but that doesn't say anything profound about the nature of objective reality, it just says that you picked a different coordinate system.