master or master's

Solution 1:

Addressing your first question:

"I was halfway through my master" is unreasonable. Your source gives options (1) master's degree and (2) Master of [insert the study of this specific degree]. Cutting those short, you could derive only "I was halfway through my master's" or "I was halfway through my Master." "I was halfway through my master" is not only made up, but unreasonable.

"I was halfway through my master's" would make sense depending on its context. I and my colleagues would understand "degree" unless your statement followed "The portions of my and my master's dinners rested warmly on the counter," e. g.

Addressing your second question:

Use neither. From your source: capitalize only specific degrees. If you do not specify, you should not capitalize.