Yes indeed green beans and fine beans are the same thing and same type of bean.
Green beans and fine beans are just a smaller variety and version of the French Bean which is the same type of bean as well.
Different parts of the world call green beans different names such as French Beans, Fine Beans, and also Bobby Beans, dwarf beans etc.
All those types of beans are the same thing except the sizes between them can be different.
Other than that there's no difference in taste or anything with the green bean vs the fine bean.
My grandma used to call them fine beans while my mother and I have always called the green beans the green beans.