Is it common to expand lecture notes for a probability master’s thesis, and how much technical detail is needed?

Standards for MS theses vary widely. What you describe seems to me to be a the low end of the scale. It might be acceptable but, depending on the institution and the advisor, it might be greatly deficient. Ask the advisor about the standard and read a few recent theses at your institution.

What to focus on depends on you overall goals. If you intend to go on to a doctorate, you need deep understanding. For a job it depends on the requirements. Some will require being able to do "heavy calculations" and some not.

But the less you do, the fewer your options will be. I suggest you raise your standards.

I can only authoritatively speak to a handful of American mathematics programs, though I do think that most American institutions are basically the same. Note that standards do vary a bit in the US, and vary widely globally.

The general sequence of higher education is something like the following (very roughly):

• the first two years of an undergraduate program lay a foundation—there are a lot of general education courses, and the basic goal of these first two years is to ensure that students are well-rounded and have the basic skills to succeed in a specialization;
• the second two years of an undergraduate program allow a student to start specializing a little—students focus their coursework on a particular field, and build up foundational knowledge in that field (in math, one would generally study well-trod topics in analysis, algebra, topology, and number theory; in anthropology, one is introduced to the four fields and studies the history of the field in general; etc);
• the first year or two of a graduate program generally consists of coursework, which is meant to introduce students to not quite the cutting edge, but to relatively recent work; this coursework is meant to put students into a position where they could produce original research, but students can "masters out", typically by completing qualifying exams and/or writing a masters thesis; and
• the remainder of graduate studies typically focus on research, with the goal being to produce something original in order to earn a phd.

The primary difference between a masters and a phd is that someone with a masters degree is not expected to have produced original work. Anecdotally, my masters institution (which did not have a phd program in mathematics at the time) allowed students to complete a masters degree by either submitting and defending a thesis, or by taking a sequence of comprehensive examinations. Most master's theses there were summaries of existing work—think "survey papers". At my phd institution, there was no such thing as a "masters student" (all graduate students were in the phd program), but one earned a masters upon passing their qualifying exams and filling out the appropriate paperwork. Again, no expectation of original work.

To answer the question: in my experience of a couple of dozen mathematics programs in the US, a masters degree does not typically require the production of new or original work. A masters thesis, if it is required (or even an option) need not be original (it could be, but it doesn't have to be). The plan outlined in the question here seems very reasonable (to me) for a masters degree.

All of that being said, one's advisor should be the authority on what is expected. If your advisor is happy, you should be happy.

In pure mathematics, it is quite common for master's theses to be surveys or otherwise non-original research. Of course, one should always aim for original contributions, even if it's just explicitly writing down folklore results. But what you are doing seems fine to me.

should I focus on understanding these technical details, or is it better to focus more on the underlying theory and main ideas, while letting go of some of the heavy calculations?

It depends on what you want to achieve. If you want to learn the technical details and use those hard analysis techniques to solve problems in the future, you can focus on those techniques. Otherwise, it might be better to focus on the big ideas. I'm not in hard analysis, so don't take my word for it; you should consult your advisor about this.