So...a couple things I ask about and noticed...
I think you are referring to a running martingale since you referenced a western training fork as similar...
Would look like this if placed on the ground..
...neck strap, strap between front legs attaching to girth and the "Y" the reins thread through.
The use of rein stops is strongly advised so you not catch the martingale rings on the bit or rein closure and get hung up and a disaster take place.
Now personally I don't like a running martingale but will use a standing martingale adjusted correctly that does not ever exert pressure to the horses mouth.
A standing martingale must be used with a regular caveson as it is slid on and buckled to it then a long strap between the front legs and attaches at the girth same as a running. It also has the neck strap to keep long straps from entangling legs in motion.
It never, ever interacts with a horses mouth, period.
It does limit the height to which a horse can extend their neck skyward.
Standing martingales caught a lot of flack because people did not use the equipment properly, but more as a head-check and very restrictive...not what this was ever intended to be.
When properly adjusted the horse has full range of motion, carrying the head and neck safely and correctly...it just when used properly keeps you from having a face of blood...but you are not engaging with the mouth, exerting forces tremendous against bar and soft tissue when working to regain control of a giraffe horse taking the bit and running...
So, a article about all the different types of martingales available, pros and cons of each... https://todaysequine.net/martingale-horse/
Cute horse to start with.
First picture who ever is riding is using a ill-fitted noseband, appears a drop noseband.
To low, it needs raised so 2 slim fingers width below the cheekbone at most.
The picture makes it look like it is sitting at a pinched angle at the bit and mouth corner.
I get the horse likes to move out when cold & crisp weather...but relax the arm and put a bend in the elbow so some forgiveness to the animal working, doing as you ask it.
I'm not positive but also think there is a strap "bubble" of improperly adjusted equipment at the cheek/jawbone.
All or any one of those things could make a horse not respond, be feeling less in control to the rider and also allow the horse to get strong and "take the bit" and go.
I absolutely can see the horse tossing their head and going...looking for the vise-grip to be released on their face..
Equipment adjusted properly is so important.
You can't go forward correcting a issue unless you first make sure all is fitting and no discomfort is happening to start with.
Your second picture...love the expression!!
So...you have quick clips on your reins and rein stops seen. Where are you placing the martingale rings at?
If those rein stops are not slid forward can guarantee you will catch those quick clip rein snaps and a catastrophe is going to happen.
Again, please please make sure all is in place at proper locations and the right equipment is used when now adding specialized "training" extras.
It takes a few minutes extra to replace our shortcut with real reins with correct closures on them...
It takes only a second to get hung up and disaster strike...
Since most use a figure-8 as a drop noseband, mouth steadying and closing device...be very cognizant of the pressure, the force exerted now if your horse giraffes and the martingale engages.
A drop noseband snugged, running martingale which just by how it is made weights the reins and adds pressure to the mouth...hands not light as a feather sending communication to the horse...
Yup, I can see a giraffe and then a bolting one running trying to get away from the discomfort and pain they don't understand why they are being given..
Make sure teeth have no issues with that figure8 noseband either and that the bit is sized wide enough and also to fit the oral groove properly since all gets compressed easily with this and if to tight...the horse can not move their jaw to soften in your hand as you seek..
With your description and comments...
I side with your boyfriend that the friend exercising Lucky is not using that martingale properly.
To me, she sounds to be using it to force a headset, to stop evading the reins and pressure exerted...
Oh yes, Lucky absolutely can evade...not only giraffe but go up and over when things escalate...flip over.
The horse tucking their face behind the vertical...someone taught that or Lucky learned to do that to get a release of pressure on the mouth and face.
I don't know how to fix it without sitting on the horse as each horse is different...you need to find the trigger for that tuck to start...
Pain, pressure...something is a trigger factor. The something is what you need to find.
Mouth check, saddle fit correctly, no pinch, no slip but good fit, bridle and bit checked for rough edges, adjusted correctly, reins secured so training equipment not get hung up on them...rider calm and relaxed, arms limber and soft following the horses motion and forgiving...
The horse...being a horse sometimes they do need to burn off some energy.
Sometimes the best thing you can do is take the horse for a ground-pounding gallop and get it out of the system...but make sure you know how to stop a headstrong horse who loves to run... or gallop in a fenced riding ring...just let Lucky stretch the legs...
That is what I thought when reading your post...
I hope you find a solution...but a martingale yes, for the right reasons and with proper accompanying adjusted tack so all are safe working their partnership together.
A lot won't agree with the standing martingale comment I made, but fact is fact.
Look at it, look at how it attaches and tell me where the mouth pressure is..it isn't.
Running...by its design and being run through the reins...when you pull back & horse raises their head...look at what engages and tell me there is not excessive force put to a sensitive delicate mouth..
When left to long a adjustment as some articles describe,...what is the point when you have no help, none whatsoever to lower the head.
All information found and discovered needs careful consideration and evaluation...
Best of luck.