MA - Master of Arts
MSocSci - Master of Social Science
MEd - Master of Education
MSci - Master of Science

The M stands for 'Masters' (so the person has a Masters degree)

The 'Education', Science, Arts, Social Science bit stands for the faculty that the person was enrolled in.

Some Universities classify Psychology in the school of art (which typically is distinct from a school of fine art - ie music, painting etc)
Other Universities classify Psychology as a science
Other Universities classify Psychology as a social science

So if the person has a Masters degree in psychology they could come out with any of the above depending on where their University has put their psychology department.

Aside from this... There can be variation between universities...

Universities with a strong (competitive) department typically require more (because of the competitive nature of the department.
Universities with a weak department (one that struggles to attract students) typically require less (because they need students to keep the department running).

IMO it is a myth that the requirements on science students are more stringent than the requirements for social science / arts / education students across the board.

It depends on the strength of the department
Which is typically reflected in reputation
For the reason that stronger departments can (thus do) require more from their students

Education...

Councelling degrees are sometimes classified as being part of the school of education.

I took a human development paper or two from over there...