How is a data set generated from a distribution? I am a mathematics student trying to learn some statistics on my own. In the context of many statistics books, the language "the data is generated from (or came from) certain distribution" appears a lot. I am trying to understand the exact meaning of this sentence. I will elborate my question with one specific example I came across in my reading.
In the book "the Elements of Statistical Learning" (ESL), the following classification example is discussed in section 2.3:

The topic of the section is to illustrate the method of least squares and nearest neighbours through this example. In the discussion of which methods is better, the auther gave two scenarios, I quote:
"Scenario 1: The training data in each class were generated from bivariate
Gaussian distributions with uncorrelated components and different
means.
Scenario 2: The training data in each class came from a mixture of 10 low-
variance Gaussian distributions, with individual means themselves
distributed as Gaussian."
I will now express my confusion using the language of scenario 1:
Question 1: By the word "data"( or "training data"), do we mean the set of coordinates that is shown in the diagram and their color? In other words, "data" refers to the set {((x_i,y_i),color_i)} that is given in the diagram?
Question 2: How is this data generated from a bivariate Gaussian distribution? My thinking is the following:
A bivariate Gaussian distribution is a function that takes two inputs (indenpendent variables) and gives an output (a scalar). So to say the data above is generated from a bivariate Gaussian distribution is to say that I put all the coordinate of the points in the diagram into the function, and use the function output to determine the color of the points (btw, the rule is that coloar blue is 0 meaning function output value less than 0.5, and otherwise orange). If this is the exact generating process, then my question is that why are the points in the above diagram so randomly positioned? Why can't we input a set of orderly points into the equation (i.e. Gaussian distribution) and color them accordingly? Say we can pick the set of points {(x,y) | x, y $\in \ \mathbb{Z}$}? (It would make the diagram much prettier at least.)
If the generating process is not what I proposed to be, then what does it mean for the data above to be generated from a bivaraite Gaussian distribution?
Question 3: Senario 2 confuses me entirely, how is one set of data generated from 10 different distributions?
It is my first time posting a question here, so feel free to tell if there is anything I did wrong or inappropriate. Thank you very much!


So to say the data above is generated from a bivariate Gaussian distribution is to say that I put all the coordinate of the points in the diagram into the function

That would imply that the coordinates were given and then afterwards put through a deterministic process. I do not have that particular book but I am quite positive that something else is meant: The coordinates of the points are the result of a random process. (And as the authors had no better method at hand they were produced by a pseurorandom process in their computers, probably some R 'function').
As I am not a mathematician I do not know appropriate notation. I'll do what we do often here, I'll take the computer language R for notation. If in R I say
MASS::mvrnorm(1, mu = c(0,0), Sigma = matrix(c(1,.5, .5, 1),nrow=2))

it will answer with two coordinates of a random point
[1] -0.09439794  1.52760301

And if I do the same again I will get two different coordinates and there is no practical way to predict those. However, if I do that time and time again the data will be a random sample from a certain distribution, in this case

To take a sample from 10 different distributions just means to perform that process with ten probability density distributions and then build the union of the randomly drawn points.
We could start producing something like the graphic above as in
blue <- MASS::mvrnorm(100, mu = c(1.5,-1), Sigma = matrix(c(1,.5, .6, 1),nrow=2))
orange <- MASS::mvrnorm(100, mu = c(0,1), Sigma = matrix(c(1,-.3, -.3, 1),nrow=2))

plot(orange[,1], orange[,2], col = "orange", xlim = c(-3, 3), ylim=c(-2, 3))
points(blue[,1], blue[,2], col = "blue")

and get

