The text of the amendment says they pick between the "two highest numbers on the list". My literalist reading of that is to pick between all 3 in this case.
However, the text for the president is a bit more confusing since it says "the persons having the highest numbers not exceeding three". To me that implies that if there is a tie for 3rd and 4th, the house is free to choose between just the top 2? And its not clear to me whether the "three" modifies persons or numbers.
I'm sure any judge would throw out my reading.
Your reading would mean that if we had:
Republican: 268
Democrat: 268
Whig: 2
The Senate could theoretically choose the Whig candidate, which clearly isn't what was intended. Yes, the 12th could use clarification.