Question

What competing views of the presidency were held by the framers?

Answer 1

Long-term v. short-term presidency: Some wanted lifelong terms down to two year terms--there were concerns of chaotic changes not allowing for stability and the states' rights side (Anti-Federalists) wanted the ability to change power often. 
There were also discussions regarding qualifications and powers of the president. Most of the disagreements were decided by compromise leading to the government that is created out of the Constitutional Convention.