Is it possible to have non-zero displacement at $t = 0$? Consider the attached graph. It shows the displacement at $t = 0$ to have some non-zero value. Is this physically possible?
This is what I think:
The displacement vector of a particle between times $t_1$ and $t_2$ (assuming $t_2$ $\geq$ $t_1$) is defined as (the position vector of the particle at $t_2$) - (the position vector of the particle at $t_1$).
In order to specify the displacement at some time $t_2$, you need  to specify a time $t_1$ (such that $t_2 \geq t_1$).
Therefore, in order to specify the displacement at $t_2$ = 0, we need a $t_1$ less than or equal to $t_2$. The only solution is $t_1$ = 0 since negative time does not exist. If $t_1 = t_2$, then the displacement is 0 between $t_1$ and $t_2$.
Therefore, a non-zero displacement at $t = 0$ does not exist.
Is my thinking correct?

 A: Yes, it is possible to have non-zero displacement at $t=0$. No, your thinking is not correct, mainly because your definitions are not correct.

The displacement vector of a particle between times $t_1$ and $t_2$ (assuming $t_2 ≥ t_1$) is defined as (the position vector of the particle at $t_2$) - (the position vector of the particle at $t_1$).

Ok, this is true, but one first has to define what is meant by "position vector at time $t$". Next, you say

In order to specify the displacement at some time $t_2$, you need to specify a time $t_1$ (such that $t_2≥t_1$).

Not true.

Therefore, in order to specify the displacement at $t_2 = 0$, we need a $t_1$ less than or equal to $t_2$. The only solution is $t_1 = 0$ since negative time does not exist. If $t_1 = t_2$, then the displacement is $0$ between $t_1$ and $t_2$.

Again not true. You're especially confused about the role of $t$ here.

I feel like in all of this, your understanding of the "origin of space" and the "origin of time" is not clear. Note by the way I'm not talking about some deep physics issue about where space and time come from etc. I'm just referring to the origin in the sense of "from where/when do we start measuring from".
For example, suppose we are all at an Olympic race track about to participate in a 100m dash. The track of course has a starting line and an ending line (where the distance between these two lines is of course 100m; and maybe for conceptual simplicity, you may want to assume there is only 1 lane rather than 8 of them).
Now if I want to introduce the concept of position vector, I need to first fix a point $O$ (called the origin), so that given any other point $P$, I can define the quantity $\vec{OP}$ as "the position/displacement vector from the point $O$ to the point $P$". In the context of the 100m-dash, a very natural choice for the origin $O$ is the "starting line" of the race.
Next, I have to introduce the "origin of time". Physically, what this means is I have to declare "when I start my stopwatch", i.e $t=0$ corresponds to the instant when I start my stopwatch. In the context of a 100m dash again, the $t=0$ corresponds to when the gun is shot, and the stop watch is started (and then about 0.2s later all the athletes start running, and in about 10s all the athletes complete the race). Does this mean that before the officials started their stopwatch, the entire world did not exist? Of course not. The world exists perfectly well even before some race officials start their stopwatch; it's just that for the purposes of the race the only thing we care about is that 10s span from when the stopwatch is started to when it is stopped (when the final player crosses the finish line).
Now, finally getting to how a non-zero displacement at $t=0$ is possible. Let's just think realistically. If I were to race fairly against an Olympic sprinter, I would lose very badly. On the other hand, suppose that the sprinter starts at the starting line (remember we defined this is the origin $O$), and suppose I started 90m in front of him. Now, suppose the race organizer says "on your mark... set... gunshot" and then starts the stopwatch (i.e $t=0$). So, you see how at $t=0$ I am not located at the origin, but rather 90m ahead of it. This is how it's possible to have a non-zero displacement even at time $t=0$. It's all a matter of where we are measuring the displacement with respect to, and when we start the stopwatch.
A: You can have a non-zero displacement at $t=0$. Both displacement and time are relative quantities. Thus, you can define what you mean by “0” in any convenient way you choose. If you want $t$ to be zero at a time when displacement is non-zero, then that’s your choice. In fact, it may be that displacement is never zero!
By the way, negative time does exist. Since time is relative, any time before your defined zero is negative. There is no problem with this. It does not imply that time is going backward. It’s just your choice of coordinates.
