We practised very basic asking her to come forward and take one step back. Today however, it took me an hour to get her to walk down to the round pen, even with the use of a carrot (normally without fail, food as a bribe always works with her).
So... she has been taught how to yield to halter pressure? Leads well for the trainer? If not, I'd get the trainer to teach HER first, before helping you learn to get her doing it.
So, assuming above is good, you ask her to walk forward, you're in 'walking forward' position(as in, you haven't turned to face her or such), you put a little pressure on the lead and KEEP the pressure on? If necessary you increase the pressure a little to make it uncomfortable for her to resist, and *whatever* she gives you, you release *instantly*(horses learn from instant association only, so need to reinforce every 'try' initially) & reward her?
Using food or other actual rewards(positive reinforcement) is great IMO. And using food as a bribe/lure *initially* in order to get a behaviour you want started in the first place can indeed be helpful. But bribes/lures do not always work, and esp if used ongoing, they tend to 'backfire' & cause an animal to do less, perhaps become untrusting, skeptical about whether it's worth taking the bribe, etc. If used, once you initially get the behaviour you want a few times with bribery, it's best to quit doing that, and just use bits of carrot(or whatever she loves) as positive reinforcement('reward' *when* you get the behaviour you want, not offered before).
She stands completely still while I lean backwards putting slight pressure on her lead rope (she has recently had a neck injury and is very sore, so I'm trying to use as little pressure as possible). This really didn't seem to work, she would only move when she wanted to and I couldn't pick up on what caused it.
If you're 'leaning backwards' does this mean you're facing her, trying to get her to come to you, rather than going 'somewhere'? How about if you are beside her, walking forwards yourself? And that way, if she is not yielding to halter pressure, you can 'back it up' by 'driving' pressure - wave a whip, the tail of the rope, your hand out behind her - to make it clearer.
Also I wouldn't be 'leaning', because with your bodyweight you will be pulling a lot, and also will make it hard for you to ensure the pressure quits the *instant* she gives you the slightest forward. Instead try thinking of it as 'taking & holding' the tension, rather than pulling/leaning against her.
When she moves forward, for whatever reason, *instantly* quit all pressure & reward her. Rinse & repeat, until she is reliably & softly coming forward when you ask, wiithout having to get stronger/drive her, before asking for a little more before reinforcing.
Towards the end, I had attempted multiple things and realised getting on her level, crouching down further in front of her, made her walk forward. But then I realised, is this like I'm submitting to her? And shouldn't it be the other way around.
IMHO best to forget about all the 'dominant/submissive' noises you have heard(not saying there's nothing
in it but there is a lot of crap & it's also just confusing the issue when you're such a beginner). Remember horses learn from *instant* association with cause/effect. Remember they do what works for them & quit/resist doing what doesn't work. So if that works to get her to come forward, by all means, start there. Ask her forward with lead pressure, then crouch down(or whatever it takes). Negatively reinforce(quit pressure) and reward her the *instant* she comes forward. Pretty soon she will be coming forward off the initial 'ask' & you can stop crouching.