Terminology regarding random sample

This question is solely about terminology.

Consider the following definition regarding random sample from All of Statistics by Larry Wasserman

If $$X_1,\cdots ,X_n$$ are independent and each has the same marginal distribution with cdf $$F$$, we say that $$X_1,\cdots ,X_n$$ are iid (independent and identically distributed) and we write $$X_1,\cdots ,X_n \sim F$$. If $$F$$ has density $$f$$ we also write $$X_1,\cdots ,X_n \sim f$$.

We also call $$X_1,\cdots ,X_n$$ a random sample of size $$n$$ from $$F$$.

Consider another sort of definition I came across many sources as follows

A random sample is a sample that is chosen randomly. It could be more accurately called a randomly chosen sample. Random samples are used to avoid bias and other unwanted effects. Of course, it isn’t quite as simple as it seems: choosing a random sample isn’t as simple as just picking 100 people from 10,000 people. You have to be sure that your random sample is truly random!

I am just asking about usage of term random sample in mathematics.

Is it a pure technical term as mentioned in first definition? If yes, then why there is a wrong usage across literature?

If no, then do we need to interpret explicitly based on the context to know whether it is a collection of random variables or set of samples(instances)?

• Your first quotation is missing things, but there is an important distinction between sampling from a distribution and sampling without replacement from a population Commented Aug 16, 2019 at 11:16
• @Henry Do you mean that first one needs more context or used the word random sample losely> Commented Aug 16, 2019 at 11:21
• I mean that the two times it says "we write $X_1,\cdots ,X_n$" it looks as if we should write some more in these particular cases as we have already said that in general Commented Aug 16, 2019 at 11:23
• @Henry Yeah, sorry for that, rectified..... Commented Aug 16, 2019 at 11:26