polia's brief sentence appeared to be a more general statement aimed at a broad readership. Many US readers would indeed equate law school with study at approximately the master's level, because it comes after the bachelor's degree, which is a prerequisite for admission, but does not take as long as an academic doctorate or require as much research as a PhD dissertation.
So I would say it depends on polia's audience. Other lawyers might correctly understand 'a master's student in law' or 'studying for the degree of Master of Laws' to mean the LLM, realizing that that's actually a more advanced level of specialization after finishing law school and qualifying as a lawyer. Though if they know what the degree is, just saying 'an LLM student' might be even clearer -- or 'studying for the LLM degree,' since 'student' is a less prestigious title and sounds a little less natural.
To my ears it would be less idiomatic to say 'a Master of Laws student,' as we don't normally use full names of degrees like that. 'Master' is correct in a context such as 'the degree of Master of Laws,' but preceding a noun it's normally 'master's': a master's degree, a master's student. (A master student would be, if anything, someone who had mastered the skill or technique of being a student, in the most general sense.)
If, however, polia is writing for the general public, then 'master's' might give the impression that a foreigner just didn't know the right word for regular law school. So it could be a good idea to rephrase in a way that explains the nature of the degree; for example, something like 'a fully qualified lawyer now completing an additional/advanced degree in a specialized area of study / in tax law / etc.'
@FloFloUSA: May I just mention a couple of other points? One very minor one: It would be 'the bulk of law students study for a JD,' since the entire expression of quantity is considered a plural entity, like 'a number of people (are).'
Also, I don't think most people would call the JD a doctorate, since that term is usually reserved for more advanced academic research degrees. That the basic US law degree happens to be called a JD and thus includes the word 'doctor' is apparently just an American quirk; the term simply represents a professional degree, like a medical doctor or a dentist. As I understand it, the equivalent of an actual doctorate in law is in fact the SJD aka JSD, or now more commonly, a JD/PhD (in a related field).
Sorry to be so detailed, but all this was interesting to me too, as I wasn't previously familiar with the LLM or other advanced law degrees.