The two events are the place and time when the light starts its journey (the first event) and the place and time when it finishes its journey (the second event). Events always have both location and time. So, the two events here would be the start of the journey (location and time at the earth) and the end of the journey (location and time at the star).
Your question about the reference frame makes no sense. For example, the earth determines where and when the first is (using clocks, etc) and then where and when the second is. The distance between them *as measured by the Earth* is x and the time between them *as measured by the Earth* is t.
If the star is at rest with respect to the earth, then someone at the star would measure the same distance, x, and the same time of travel, t. The star and the earth are in the *same frame* because they are at rest with respect to each other.
Once again, it seems that you don't quite get what a reference frame is. ALL that is required to be a reference frame is that you not be experiencing acceleration. So the sun is very nearly in a reference frame (the only acceleration is due to its motion around the center of the galaxy, which is a very small acceleration). The Earth isn't as good because of its much faster motion around the sun, but can be used in many situations (and we will, for convenience).
So, two different reference frames are not accelerating with respect to each other: in other words, they are moving with some constant speed with respect to each other. The speed is called the relative speed.
Each reference frame can use its clocks, measuring sticks, trigonometry, etc to determine the locations and times of *all* events. So, the earth can measure when a beam of light leaves the earth and also when it gets to the sun (or star, if you prefer). Those are two events. The distance between them *as measured by the Earth* and the time between them *as measured by the earth* are the distance and time between them in the reference frame of the earth. So, *in the frame of the earth*, the distance between the earth and the sun is 93 million miles and the time it takes for light to go from the earth to the sun is 500 seconds. Those are measurements *in the reference frame of the earth*.
Similarly, the earth can measure the distance to a star and the time it takes for some spacecraft (or light) to go from the earth to that star. Those are measurements *in the reference frame of the earth*.
Now, one basic aspect of reference frames is that you are always at rest in your own reference frame (by definition of what it means to be your reference frame). So, if I am traveling past the earth, and someone on earth measures me to be going at 50,000 km/s, that is my speed in the reference frame of the earth. In my own reference frame, I am *at rest*. The earth is going past *me* at 50,000 km/s.
So, suppose we send a beam of light from the earth to the sun. In the reference frame of the earth, the distance the light travels is 93 million miles and the time it takes is 500 seconds. Two events: one on the earth when the light starts out, the other on the sun when the light gets there, separated by both distance and time.
Now, from the reference frame of someone going past the earth at .5c, the distance and the time of that journey of the light will be different.There are still two events: the place and time when the light left the earth and the place and time when it gets to the sun. But the *distance* between those two events will NOT be 93 million miles. And the time it takes for light to go between those two events will NOT be 500 seconds. If that someone is moving towards th sun from the earth, then the distance between those two events will be about 53 million miles and the time will be 288 seconds.
The speed of light is still the same in this other reference frame (186, 000 miles per second). The other reference frame can still measure distances and times for *all* events. But the measured distances and times will be different than those measured by someone on the earth (because of the relative speeds).
The events themselves are NOT in a specific frame: it is the distance and time between those events that are in some frame. ALL frames can measure distance and times between those two events. ALL frames can compute the speed of light by taking the distance *as measured* by the time *as measured* and all will get the same answer: 186,000 miles per second.
Similarly, *every* reference frame will compute the same proper time between two events. They *all* compute that proper time by doing
sqrt( t^2 - x^2 /c^2 )
Each frame has its own values for x and t, but ALL will obtain the same value for the proper time. PLUS, that proper time is what the frame measures for the time that sees itself as *at rest* while going between the events. In a frame that measures x and t above, the 'rest frame' is moving at v=x/t.
