I am creating a Tiefling subrace for my campaign focusing on Cold instead of Fire for their heritage. I had previously created another subrace that focused on Lightning, and that one doesn't have this issue, but, when I created the Cold subrace all of the previous traits, such as "Hellish Resistance" and "Infernal Legacy", still remain, and have the effects that come with them.

From what I can remember, I haven't built the subrace any differently to the Lightning based subrace.

Does anyone know why it is doing that, and/or have a solution?

Do you mean variant race? You can post a link to unpublished homebrew for mods to check.