It's hard to answer the question from the two sentences quoted. But @yunus has done well to give the source link. The article is an analysis of Britain's apparently low unemployment rate, and this is what the first section says in its entirety:
How many people are unemployed?
Officially, about 1.3 million people in the UK are unemployed.
That's 3.7% of the working-age population (16-64 years old).
[graph shown here]
But that figure represents only a small part of the 10 million working-age people who aren't in a paid job.
Nearly nine million of them aren't called "unemployed". That's because they're not actively looking for work, or available to start a job.
Instead these people are called "economically inactive".
In fact, more of them say they want a job (1.7 million people) than are officially unemployed.
So the whole point of this section is to get a better understanding of the terms used. For instance:
Unemployed = "Officially unemployed" (paragraph 1). (Perhaps this means that these are people registered with the government agencies as looking for work.) The number is 1.3 million.
People who aren't in a paid job (paragraph 3). The number is 10 million. This is a global and non-technical description, and is intended to lead into a discussion of why the two numbers are different.
Economically inactive (paragraph 5). This number is "nearly 9 million". This is the group of people who are working-age (paragraphs 2 and 3) but "are not actively looking for work, or available to start a job." (paragraph 4)
People who say they want a job (paragraph 6). This number is 1.7 million.
So how do these terms relate to each other? That's actually not an easy question, and I suspect that's one purpose of the article: to show that when we dig deeper, there are different nuances to consider.
I think it's unclear from the wording how the 1.7 million is counted. Is it 1.7 in addition to the officially unemployed? (If so, the total who want a job is presumably 1.7 plus 1.3 equalling 3 million.) Or is it 1.7 including the officially unemployed? (In which case we would be adding 0.4 million to the official unemployment number.)
One final note about context. The graph in the middle of the section splits the text in half. All the text quoted in the OP comes from below the graph. But the paragraphs above the graph help with the understanding. Sometimes when reading a multi media document we need to work harder to see which parts of the text are relevant to the interpretation.
B. The question
You are making a correct deduction when you say that some of the "economically inactive" group want to work, and some don't.
You are not making a correct deduction when you say that this number is rising. All the numbers given are one-off or point-in-time numbers. The definitions are intended to make comparisons between groups, not comparisons over time. All we can say is that one number is bigger or smaller than another number now.
You are making a correct deduction that "this number is greater than official statistics show".
Why are they called "economically inactive" if they are looking for a job? Good question. Answer 1 is to note that the reasons why people are not working are many and varied, and they are in fact explored in section 2 of the quoted article. Answer 2 is to give a small personal example. I have a family member who would be economically inactive at the moment, but still wants to work. He's in both of those groups because he is currently recovering from a serious injury. We hope that in time he will be back to full health and productively employed.