Since I train judo as well as Brazilian jiu-jitsu and the occasional wrestling class, this comes up a lot in my own training.
Judo's morote gari (two handed reap), wrestling's double-leg takedown, and BJJ's double leg takedown are all the same technique. Except not. Just like the similar-but-not-the-same techniques you found shared between Hapkido and Aikido, each one has just enough individual detail to distinguish itself from the others.
In judo, one wants to avoid initiating groundwork, so wrestling's knee-slide footwork must be kept to a minimum. In BJJ, one must protect against the guillotine choke, so head position is changed and the chin must be kept down. In wrestling, the head is used for an extra few percent of power and control, so the chin is kept up and the head is pushing hard. These are zero-sum differences. One cannot do the wrestling version optimally if held to the judo standard, and thrice-versa.
These changes do not exist in a void. They have a purpose. Judo's version avoids wrestling-style footwork due to a rule in judo competition: a throw does not count for points if the thrower first goes into groundwork. BJJ's version sacrifices power and control in order to avoid being choked. So I have a clear choice before me: am I worried about getting choked? Do I care about judo's rules? I can determine which version I should practice based on my own priorities.
This is the missing step in your question. Maybe we don't want to excise our old habits. To take an example from my old karate school: teaching a student the Isshinryu method of a spinning back kick will make them needlessly worse if they already have an awesome Soryu spinning back kick. It's better to leave the kick alone.
We should be careful of the traditional approach that says "whatever we teach here is the best". We should be open to different approaches, to new approaches, to foreign ways of doing the technique. (Whoever runs the class obviously has the final say over what gets taught, what gets practiced, and what is safe for sparring in their class. But we shouldn't do what we do just for inertia's sake.)
How to work around interference
If one does make the decision to break their old method and imprint the new one, they should understand that it will require them getting worse before they get better. One must pass through the valley to get to the summit.
Such a student should be taken on careful and slower road to learning the technique. More drilling, more break-downs of individual steps of the technique, and more explanation is called for than with a brand-new student.
At first, they should be given fewer high-stakes opportunities to apply the technique quickly and naturally. If most of the training is in full-on sparring, the wrestler will keep rolling the dice with their wrestling double-leg instead of practicing their (worse) BJJ version. A higher volume of lower-intensity sparring is called for in this instance.