As you have noticed, mathematical text is often quite concise and it can be difficult to squeeze it into tighter space or write in your own words in a nontrivial way.
Therefore it is easy to end up copying things verbatim if you want to take notes.
Taking useful notes in mathematics is quite different from many other subjects, and you are not alone with your problem.
You should not be embarrassed; you recognized a problem and want to do something about it and far too few students actively do this.
I have some suggestions for taking notes in a more useful way.
(This is, of course, based on personal experience and opinion.)
Reading a book and following lectures are two different ball games.
Let me start with the lectures.
It often happens that lectures move on so fast that if you want to copy everything that happens on the blackboard, it can be difficult to follow the content and — most importantly — the ideas.
In these days lectures should not be the sole means of delivering the content; the professor can scan his notebook and put the pdf file online for the students to download.
If he does not do this, request that he does.
Mindless copying is pointless and wastes a lot of time.
(Arguing this to some professors can be a difficult task, but it should be tried.)
If the lectures are based on slides, ask for them and bring them with you to the lectures, printed on paper.
If no such material is available, try to read the textbook or other such material in advance, so that you will have the material in mental form.
I have found it very useful to have the lecturer's notes he is using (or very close to it) during the lectures.
This does not mean that I sit down idly.
I follow the lecture and try to read the material in advance (at least quickly).
There are always some tricky points that are not perfectly explained in the written notes.
Often the professor says helpful remarks and answers questions from the audience, explaining what is happening or why certain choices were made.
It is those gems that I try to catch and write them among the lecture notes.
The notes can be informal and do not attempt to make rigorous mathematics.
Instead, I try to capture my momentary intuition in words and explain things to myself or relate them to other things.
The kinds of notes I take depend heavily on my proficiency in the subject and the notes can be of any kind.
(Sometimes I just keep reminding myself about the nature of each object by comments: this is a function, this is a vector, this is a set of sequences…)
The notes are rarely so lucid and well explained that no notes are needed, but this can happen, too.
Most of the time I just follow the professor's train of thoughts and try to understand why everything happens as it does.
Only a small portion of time is spent on making notes.
If I have to take notes to copy the content to my notebook (this is sometimes the only way to get it, unfortunately), I have much harder time understanding.
If you are reading a book, the situation is different, as there is no hurry.
Making supplementary notes works here like during the lectures.
If you don't want to damage a book with your own writings, I would suggest post-it notes — I at least give great value to having notes right next to the content they comment.
My notes are mostly comments.
I find copying theorems and proofs verbatim quite useless.
I would suggest reading the theorem and its proof in detail and then trying to figure out what it is about, what it means.
Make sure you understand the overall plan of the proof, not just the technical details.
If you want to take notes, put the book aside and write the theorem.
If you understood the statement and the underlying machinery well, you should be able to write the statement.
Then write a proof, at least a sketchy one.
If you get stuck, look at the book; it doesn't matter if you have to look every three lines, as long as you really try to do it yourself first.
Now you have notes about the theorem, and you have written them down in your own words and as your own thoughts.
It might be almost exactly the same proof as in the book, and that's ok.
It came from someone else, but it's your proof now.
If I have to choose one of your strategies, I would go with 3.
Do what feels comfortable and natural to yourself, but make sure there is a point to the notes you are taking.
There is one more thing that I must add.
Doing exercises is very important in learning mathematics.
(I don't know about microbiology.)
Do them, and if there are none, make some up and do them.
I think the most important notes I have taken in my years of studying mathematics and theoretical physics are solutions to exercises.
Exercises give you hand-on experience of the power of your new theorems and make you comfortable with the machinery and language of the theory you are studying.
Something of a summary:
Take notes only when there is your own thought in them.
If you find yourself copying without thinking, there should be another path to the bulk of content.
Use notes to explain things to yourself, not to build all of the theory.
Let me elaborate on making up your own exercises.
(I did this in the comments, but it deserves to be in the actual answer.)
If you are curious enough, you don't need to produce problems artificially.
Try to dig a little deeper than your course.
How are different things related?
For example, is the sum of injective functions injective, or is a connected set always measurable, or is every square root of a continuous operator continuous?
The questions can be anything, and they can be silly.
Is a smooth function a Banach space, or is a finite group measurable?
Learning what things are completely unrelated and what things don't even mean anything is useful.
I find this kind of mental gymnastics entertaining and it gives me a better touch to the subject.
If no questions occur, another good source of questions are the main results of your material.
Can you write a list of them without looking?
Can you prove the things on your list?
Can you define all relevant objects?
Do the theorems remain true if you assume a little less?
Why not, why yes, or why is it hard to see with the tools you have?
In my opinion, informal exercises are important.
The problems in exercise sets in books, courses and exams tend to be quite formal, but that doesn't mean that only formal mathematics is good mathematics.
Play with your new tools and you'll learn to work with them more efficiently.