Well not being a shirt and tie wearer, I don't know if I am qualified from experience to answer this, but I do understand English, so hopefully the meaning is clear enough! Snugly does indeed mean closely fitting (in this context). So the tie should be snug and firm around the neck, not loose, but not so tight as to cause discomfort. To button the shirt closed, merely means to do up the top button that holds the collar closed. I think the word 'closed' here is unnecessary, as to button a shirt, means to do up the buttons. To undo the buttons is to unbutton. When you do up the buttons you have no leeway in how tightly or loosely the shirt is closed, unlike the tie which you can pull tight, or loosen, so you are right, that the 'closed' when referring to the shirt doesn't carry any connotation of tightness or snugness - and as I said, is probably unnecessary anyway.

Snugly means not loose. Not tight, but just a nice fit where it looks neat. To have a shirt buttoned closed means that the shirt must not be open. In other words, all the buttons must be fastened. Try not to confuse close (opposite of far) This has a hard S sound with close (opposite of open) :) This has a soft Z sound.