A lot of my consciousness of gender-free English calling has come from callers who do something different from what I expect, so this is very much about Boston GF calling. There are some smallish differences in how it's done in other parts of the US—I'll try to point those out below. Some of these descriptions make use of landmarks in the Jamaica Plain hall: there's a big clock on the left side of the hall, and a wall of windows on the right side—translate as needed.
This style of calling, while it's not 350 years old, is 35 years old—there is a real tradition here. New-to-GF callers who decide they prefer something else (which is different from making mistakes, which we're pretty forgiving of) should know that it'll come across as disrespecting that tradition.
Calling is based on current position. "Original first diagonals" shouldn't be used (people don't remember and have to move themselves back to that location in their minds, which means it takes too long to figure out). "Left file" means current left file, not original, etc.
The exception to that is 'ones', 'twos', 'threes'—those are roles, not positions. "Current ones position" will be confusing (or at least unexpected). We'd call that "ones" (or "twos") or (maybe a little more often) "top dancers". If it's a mix of ones and twos (e.g., after people have moved three places around the circle), we'd definitely say "top dancers" ("middle dancers," "bottom dancers").
If you've got nameable landmarks in the room, use them (clock side, window side; lake side, field side; etc.) rather than left file, right file. It's often easier/quicker to remember, and won't be confused with other "right" or "left" calls (turn single right, left-hand turn, etc.).
When three people in a set of four are doing something, it's usually (though maybe not always) better to use "pair plus one" designations, rather than three individual designations; e.g., "ones and the second window person" rather than "first clock, first window, second window," or "first diagonals and second clock" rather than "first clock, second clock, second window." If the previous (or next) thing that happens involves two of those three, they're who I'd name as a pair (e.g., "first diagonals and second clock" vs "clock dancers and second window" vs "twos and first window").
In Boston, we would say "end crossed over" rather than "end improper." Because so many of us also do contra, "proper" and "improper" are thought of as formations of a whole line, not positions of individual dancers. "End improper" will therefore lead people to think about whether they're a one or a two and whether they're in top position or bottom position, before they decide which side they're supposed to be on (and if they're a two in top position, they won't be able to decide). Similarly, if the twos are below, "twos end improper" and "twos end proper" would lead to the same positioning. "Crossed over" is unambiguous. (In other parts of the country, "end improper" and "end crossed over" are used as synonyms, so this is a Boston thing, or a lots-of-contra-dancers thing.)
"First diagonals" is sometimes also called "right diagonals," but the latter isn't language we use in Boston, and that they're the same thing isn't obvious: when you're interacting with your diagonal, you're facing into the set of four, and so that person is right in front of you, not to your right. I do think this is a concept that could easily be learned (i.e., when you're facing across the set, first diagonals are right diagonals and second diagonals are left diagonals), we just haven't taught it that way in Boston. This is definitely a good way of thinking about it in a set of four.
Because we haven't used right diagonal to think about first diagonal in Boston, the concept is very difficult for us in a square set; the first diagonal goes the "wrong" direction (for head couples, NE to SW rather then NW to SE). And how far right do I look to find my right diagonal in a set of eight? There aren't a ton of square-set dances in ECD, and fewer still that call for diagonal interaction, so Boston GF dancers will need pretty explicit explanation.
The tradition for finding partners is to just get in line, and your partner is the person you're across from. Hence, callers don't say "Find a partner and join the set." (Maybe "Join the set to find your partner" instead.) If there's someone you really want to dance with, you're encouraged to join the line at the bottom after the set has formed. Our tradition is also to not have a waltz—the emphasis is on the group dancing together, not individual couples.
In duple or triple minor dances, we rarely worry about proper vs improper dances, and the caller rarely mentions it. (In improper set dances, where dancers end on a side different from where they started, it's helpful to let dancers know that's what's supposed to happen.)
I note on The Heather and the Rose website that they decide who goes first (e.g., in a figure eight) by saying that it should be as though you're passing right shoulders (assuming the choreography doesn't more appropriately have you doing something else; e.g., if one of you has to do something next, that's who should usually go first). We've never made a rule about who goes first in Boston (though I wouldn't mind adding it).
Something you might be interested to know, but definitely don't need to teach (because we don't really pay much attention to it): the gender-free hand-holding is left hand up, right hand down (though that might change with, e.g., a gate turn). That's east coast. On the west coast, it's left hand down, right hand up. (Yike!)
A couple of words on philosophy. What's probably obvious is that gender-free terminology is more inclusive. For new dancers, it can be simpler and clearer (certainly clearer when people are dancing the "wrong" role). It also usually requires fewer words. More basically, we think that teaching and calling based on position rather than role makes it easier for dancers to see the pattern of the whole dance, rather than what they as individuals are supposed to do. To the extent men and women have different roles in English dance (thought that's much less than in, for example, contra), knowing both roles makes everyone a better dancer. It's also our experience that long-standing gender-free dances feel like a group of people who come to dance together; at long-standing gendered dances, it feels more like a group of men who come to dance with a group of women.