Double proficiency from blessing of knowledge

A player in my new game chose knowledge domain gaining blessing of knowledge.

At 1st level, you learn two languages of your choice. You also become proficient in your choice of two of the following skills: Arcana, History, Nature, or Religion.

Your proficiency bonus is doubled for any ability check you make that uses either of those skills.

My player chose two of the skills but had the other two from elsewhere and thus noted double proficiency in all four.

I’m pretty certain that double proficiency is only gained on the chosen two skills, because

  • It was implied in this question: How does the Cleric's Blessings of Knowledge feature interact with the rules for gaining proficiency when already proficient in a skill?

  • It seems like in 5e you do not usually gain double proficiency for skills you might not even be proficient in. I might be wrong on this one.

  • It says "either of those" instead of "any of those" implying that it is of two and not of four. Then again, I’m not a native speaker.

I told the player my reading of the feature and they accepted it without problem, however, they said they find the wording ambiguous.

My question is: Are my assumption above correct and can blessing of knowledge give double proficiency in skills that you might or might not have?