What would be the proper way to ask for reference letters So, during my bachelor years I was a bit of a hermit. I studied on my own, made little contact with my professors outside of the necessary academic/ social situations, and mostly passed on as that rather good student professors don't think much about.

Stupid, I know, but people skills is the worst of my worst developed skills. Also I was stubbornly independent and asking questions during office hours made me feel as if I was failing at being a decent student, so I never asked questions or help.

4 years after graduating, I am thinking of applying for a master degree. I have picked the professors to whom I will ask for reference letters .... and the first deadline is 31 jan. I have received BA and BB with one of the professors (professor X) and BA and AA with the other (professor Y). 

I was thinking of emailing professor Y and asking if I could meet him (it is easier for me to ask for the letter face to face, and I think I can make a better case this way). He is a positive, decent person and I think he considered me a good student despite our minimal interactions due to the cold and introverted disposition we both have as people. Also, knowing how busy he usually is, I don't intend to take much of his time and I intend to clarify this in the email.

As for professor X, I will just email him. He is very indifferent towards all students, me included, so nothing I can say face to face will make a difference. The problem with him is that, he probably doesn't even remember me. 

So I am not sure if I am going about it the right way. The reason I left this for the last moment is that whenever I opened my email, I felt there was no right way to do it. What would be the most appropriate way to ask for the two reference letters?
Dear Professor Smith:  

My name is Freya ASE.  I was a student in your class on widget design at Enormous State University in 2014.  I am writing to request a letter of recommendation.

After 3 productive years in the workforce at Ballbearings Anonymous, I have decided to apply for the Ph.D. MechE program at CalTech and a few competing programs.  While BA has been great practical experience and good use of my ESU training, now I want to move into cutting edge fluid flow research.

Would you be able to write a positive letter for my applications?  If helpful, I can supply some bullets to refresh your memory on my class project, class performance and our interactions.

Sincerely,

Freya ASE