A physical quantity is introduced by its operational definition. In general relativity we use a differential manifold to describe the 4-dimensional space-time and, to identify a point in it, we use a reference frame. This frame consists of an origin and four coordinate, time and the three spatial coordinates. My question is:"If we want to practically identify an event in space-time, how we measure its space-time coordinate, especially time? With a clock in the neighborhood of the spatial location of the event? With the help of some particular "game" of light signal?
A physical quantity is introduced by its operational definition.
Yes. Excellent. And physical quantities would include facts like whether an observer receives one signal between receiving two other signals or sees one mark between two other marks (clocks and rulers are designed on these principles)
In general relativity we use a differential manifold to describe the 4-dimensional space-time [...]
[...] and, to identify a point in it, we use a reference frame.
No. The manifold ready has points, which in relativity are called events. You can label them any way your want. Just like you could give observers names like Alice or Bob. This can help you to communicate. But just saying the label by itself to someone that doesn't know how you assigned the labels is not physical.
This frame consists of an origin and four coordinate, time and the three spatial coordinates.
That sounds like a global frame in special relativity.
If we want to practically identify an event in space-time, how [can] we measure its space-time coordinate, especially time?
Coordinates are not physical by themselves. If you looked at two points you can't tell their x coordinate, maybe one of them is the origin, maybe the other is, there are many possible coordinate systems. It's not an operational definition.
You could ask about the so called interval between two event along a path, and if the events were infinitesimally close you could talk about the interval between them without having to specify a path.
But you can't look at the physics and from the physics figure out something with arbitrariness like what the coordinates are in some coordinate system.
This already happened in regular Newtonian mechanics, the origin could be anywhere, the x axis could point in any direction.
With a clock in the neighborhood of the spatial location of the event? With the help of some particular "game" of light signal?
That can help you to find the interval between two events. It still can't tell you if one or the other event is the origin in some arbitrary b coordinate system since there are coordinate systems where one is the origin, others where the other is, and yet more where neither of them it, even coordinate patches where no event is the origin.
Remember in Newtonian mechanics, when the origin could be anywhere, and the x axis could point in any direction? In special relativity, the origin can be any event and the time axis can point in any timelike direction (any direction that is tangent to the world line of a massive observer or equivalently that has a tangent with a timelike sign for the squared interval).
This carries over to general relativity, you don't have global frames (that's what the general in general relativity means) but it is still the case that the tangent to an observer (no matter how quickly it moves at a sublight speed, or which direction the velocity goes) can be a time axis for a local coordinate patch.
So given an event it could be the origin of a local coordinate patch, or not. So its coordinates could be zero ... or not. And it could be the origin in some coordinate patches and not the origin in others.
And coordinate differences are not physically meaningful. In fact you write the metric in a coordinate system because you can use it to compute physical things from the coordinate system.
So the physics is "Coordinate system + Metric in that coordinate system = Physically Meaningful results."
What kind of physically meaningful results can you get? Given a coordinate path (a parameterized collection of coordinates) you can use the metric to find out if path is a possible path of an observer by seeing if the interval of the tangent to the curve is timelike (metrics allow you do compute intervals). If it is, then you could break it into regular pieces each of which has the same arc length where arc length is measured, not by coordinate differences, but by the interval you get from the metric.
Now you can relate this to clocks. If a clock travels that path it will tick an equal amount on each of those paths.
So now you can predict how many clocks ticks happen between the event where your path intersects two other events. That's something physical. And that prediction comes out the same no matter what coordinate system you used.
If you use a different coordinate system all the coordinates are different, and the metric possibly has different values but you compute the interval between events in the same way. So you will agree whether the tangents are timelike and you will agree about which events break a given lath from event A to event B into 2,3, ... or 100 pieces of equal arc length when arc length is measured by the interval.
So you agree on the physics, regardless of your coordinate system. That's great. But it means there isn't an experiment you can do to find out "the" coordinates of an event because they aren't unique or physical.
What you have to do is make a model. You make a mathematical thing and then you assert that certain parts of it correspond to things in reality that match up then you look at other parts of the model and extract your predictions from them. The coordinates you use (if any) in your model are irrelevant. Sure you might use ones that feel very intuitive to make your life easier, but that doesn't make them physical, just convenient for you.
And I said "if any" because even the use of coordinates at all is not required. All you really want to know is whether an event happened between two particular clock ticks or a ruler has two particular marks on either side of an object. Those are the kind of things you have in the lab. If you actually put the ruler and clocks into your model then that information is already there and you don't need to add a coordinate system.
What general relativity does is restrict which models you can make, it restricts you to only making models that satisfy Einstein's Field Equation. Once you have one of those, you don't need to pick a coordinates system if the model already has the information you need.
There isn't a simple answer to your question because, well, it depends.
In the lab the only time I can think of that this issue arises is with gravitational time dilation e.g. experiments with atomic clocks or GPS satellites. In that case I would guess we'd use EM signals (light or radio) and correct for the known travel time.
In astronomical contexts the only obvious example I can think of is gravitational lensing. In that case we don't observe the trajectory of the light directly but instead calculate it by observing how the light reaches us.
Observations of distant events are always a bit iffy in the context of general relativity. We can assign any event to a spacetime point in our coordinate system, but the physical significance of doing so is open to question. Only local observations are entirely unambiguous.