People in your hunt do wear a frock or hunt coat. There are basically only two types of coats worn in foxhunting. (I won’t go into the shad/weaselbelly thing)
The one commonly referred to as a “Melton” is a black standard 3 button hunt coat (same style as hunter/jumper). “Melton” is actually a type of fabric, not a style of coat. I would hazard a guess that most of your members in an English riding habit are wearing a hunt coat, and in your climate probably a light or medium weight Melton fabric; or perhaps a synthetic fabric. (I think you have mild winters, right?). It also has pockets on both sides.
A hunting frock is a different style; a different cut entirely. They are worn by members who have earned colors. (I think this can vary so others may want to correct me on that point. They are much more expensive (around here a new one can sell for 500-600$) and can be found in a cavalry twill fabric or different weights of Melton fabric. They can also be found in scarlet for staff or gentlemen members. Black or navy blue for ladies.
If you want an easy way to tell, check out the waistline. If there is a seam around the waistline of the coat - it’s a hunting frock. There is a pocket on the right (outside).
Ratcatcher is an entirely different riding habit, and is worn for cubbing only. This is usually a lightweight wool hunt coat in a tweed or patterned fabric in autumn colors. (browns, golds, greens, etc.) The jacket of this habit is also worn to hunt breakfasts or teas after hunting, as it would be rude to wear your mud splattered frock or hunting coat in to someone’s house. (you also remove spurs and clean your boots off)
This is just a nice party. Wear a nice sporty blazer, slacks, shoes, and a nice tie. Like a really nice dinner party. Have a great time.
If you want to see what different cuts of coats look like, here is a link:
http://www.foxhuntingshop.com/catalog/Coats_for_Foxhunting.html
It’s very simple.
A black or navy hunt coat for the formal season, a frock for staff (scarlet) or members with colors (black or navy) for the formal season, and hacking jackets/tweed hunt coats for all members for cubbing.