Simultaneous is a relative term

Suppose you see a person, A, standing on your left side and another person, B, standing on your right.  The distance between you and each person is equal. They both hold a flashlight in their hands. At some moment, you saw both of them turn on their flashlight simultaneously. To you, event 1(person A turning on his flashlight) and event 2(person B turning on his flashlight) happen at the same time because the distance between you and each distance is equal, which means that the light signal of both events travels an equal distance to reach you. Therefore, these two events are simultaneous to you. Suppose I was standing still relative to you, you will assume that I saw both events happen at the same moment. In fact, I did see both events happened at the same time. But, what if I was moving? Suppose I was moving at a speed of 0.5C (0.5 speed of light) toward the person A. When I was in the middle of the two people, they turned on their flashlights. In your perspective, you think that I saw event 1 happened before event 2 because I was moving toward the light signal of event 1and caught that signal before I received the light signal from event 2. That is to say, two events are not simultaneous to me, which proves that simultaneous is a relative term. Everything works perfectly so far. However, things get tricky when we move to my perspective. According to the special relativity, light travels at a constant speed in every inertial reference frame(the reference frame that travels at constant speed). To me, the light signal from both events travels at the same speed. Two flashlights were turned on when I was in the middle of person A and person B. In my perspective, I think I was standing still, is you and the two people were moving. When the distance between me and the two people is equal, the flashlights were turned on, so in my perspective, I will see two sources of light that are equally far away from me locating at the opposite direction shooting light to me. As I mentioned before that two light signals travel at the same speed toward me. Since two signals travel an equal distance to reach me, I should see two signals at the same time. That is very bizarre. Why we get different results when we change our perspective. Which one is true? The truth is, one of our reasoning is incorrect. When we try to figure out which events happened first in my perspective, there's a detail that we miss. The reason why we get a conclusion that I saw both events happened at the same time when we look at the whole thing in my perspective is that we assumed I saw both signals were sent at the same moment. It is true that both light signals travel at the same speed and travel an equal distant to reach me. However, to me, both light signals were not sent simultaneously. To you, on the other hand, both signals were sent at the same time, but one of them hit me first. Therefore, we both agree that one of the events happened before the other to me.

In the real life, we rarely disagree the simultaneity of two events. That's because even we have relative motions to each other, the speed we move is very slow compared to the speed of light. When our relative velocity is much smaller than the speed of light, the light signal of each event reaches us almost at the same time. The difference is too small that we can ignore it. When our relative velocity is very high, like 0.5C, however, the difference is very big that we can not ignore it anymore.