Suppose I analyze text and find that there are:
Are these actually statistics of the text? I am not sure because statistics involves "collection of data" but these are just "counters" of some variables. I have also seen similar examples in other fields like games where you have "game stats" which are also simply "counters". How are these counters "statistics", wheres "collection of data samples" in this?
Such counts can be regarded as statistics in both an informal and a somewhat formal sense.
There may well be some question as to whether this can in any reasonable sense be called a random sample of some population (and so whether analyses that rely on it being one can reasonably apply), but it's definitely collecting data, and fits your definition of statistics.
For example if there was interested in the average number of words per paragraph in some body of work, the information in this passage of text would potentially be relevant.
Statistics has many meanings. This doesn't often bite.
Informally, statistics can mean data, as here, or when there is an announcement of the latest trade or production statistics, or in any field when people talk about looking at the statistics. Also, one person's data are someone else's results, and vice versa. Are batting averages data or statistics? They could be either or both.
The etymology of data as Latin for (things) given can be surprisingly pertinent: from your perspective, data are what are given to you. If I work with mean temperatures in several months at several stations from an internet source, those aren't the original data, but they are mine.
There is a strict sense in which a statistic is something computed from a sample. A sample mean is a statistic. A sample correlation is a statistic. I am happy with statistics being in various different spaces, so that a sample scatter plot is also a statistic. (This doesn't always reach even introductory texts, but better texts explain the difference between statistics and parameters, or at least try not to undermine what some readers will learn later in intermediate or advanced courses.)
Statistics is also the field or discipline covering management, analysis and modelling of data -- in which we work, some of us happily and much of our time, and others only grudgingly and fleetingly.
In my view any distinction between counting and measuring is irrelevant here. Broad sense, measurement includes counting. Measuring a death rate depends on counting a numerator and a denominator. Even if anyone wants to insist that counting is not measuring, that is not material, as the field of statistics still covers both.
Adding to the great answers, a statistic is some numerical characteristic of data. (Or, a result of a function that takes your data as input). So, yes, those numbers are technically statistics.
