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For time-like separated events, reading of 2nd event of any observer from central event (i.e. at origin)

  1. should lie within top light cone (we will consider only future events and ignore the bottom cone for this discussion) and

  2. has to be on hyperbola within the top light-cone.

Now, since hyperbola cuts $t$-axis only at one point and that is the only point where events are happening at the same place, question is can there be more than one reference frames for which 2 time-like separated events happen at same place?

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Sure. Find a frame where they are at the same spatial location. Then any frame that differs by a spatial translation, a time translation, or a spatial rotation will still have them at the same spatial location.

there is a unique reference frame in which two timelike-separated events occur at the origin, up to rotations of the spatial coordinate axes.

Here they forgot time translation, but otherwise they covered it succinctly.

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Technically, yes, more than one such frame will exist. Specifically, two reference frames can have zero velocity relative to each other, but their origin points can be offset from each other, and their spatial axes can be rotated relative to each other. In such a case, both reference frames will observe the same spatial separation between any two events.

The more accurate statement would be something like "there is a unique reference frame in which two timelike-separated events occur at the origin, up to rotations of the spatial coordinate axes." In most pedagogical introductions to special relativity, these subtleties are defined away by assuming that the spatial axes of all reference frames point in the "same direction" (appropriately defined), and that the origins of the reference frames we are considering coincide at a particular time ($t = t' = 0$) as measured in both frames.

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