I am not sure about your mathematical background, but you can try "How to Prove It". I am currently studying this book myself. As someone with no formal mathematical education, and someone who had been really terrified of mathematical proofs before, I think this book is exteremely well written.

Excerpt from the book's introduction:

The book begins with the basic concepts of logic and set theory, to familiarize students with the language of mathematics and how it is interpreted. These concepts are used as the basis for a step-by-step breakdown of the most important techniques used in constructing proofs. The author shows how complex proofs are built up from these smaller steps, using detailed 'scratch work' sections to expose the machinery of proofs about the natural numbers, relations, functions, and infinite sets.

I believe it may be suitable for you, because:

When you come to an exercise, you know that you are ready for it. There is no doubt in the back of your mind that "maybe I haven't read enough of the chapter to solve this exercise"

As I've just pointed out, I don't have mathematical background, or put it bluntly, I'm quite bad at math. However, even for me, it is extremely easy to follow everything author says.

It makes it difficult to be a passive reader

Indeed it is. Besides having plenty of exercises after each chapter, there are a lot of them scattered within each chapter. Unless you devote you time and energy and solve each exercise yourself, I believe it will be pretty hard to follow anything.

It makes you become invested in the development of the theory, as if you are living back in 1900 and trying to develop this stuff for the first time

As you can see from the name of the book, the author's aim is teach students how to prove things. And, when trying to prove something yourself, you will definitely need to use your own reasoning and develop your own approaches to the problem.

You can check out the book here