How many US Presidents were elected/appointed under a different name than they were born with?

I found six and they are: Grover Cleveland-born Stephen Grover Cleveland,Woodrow Wilson, born Thomas Woodrow Wilson, Calvin Coolidge, born: John Calvin Coolidge, Dwight David Eisenhower born:David Dwight Eisenhower, Gerald Rudolph Ford, born: Leslie King Jr., and William Jefferson Clinton, born William Jefferson Blythe.

Also, Ulysses S. Grant, born Hiram Ulysses Grant.

Jimmy Carter, the name he was elected and sworn in as, was christened James Earl Carter, only slightly different but different none-the-less.

https://en.wikipedia.org/wiki/Jimmy_Carter

A lot of men go by their middle name rather than their first name; it's not a "changed" name. Excluding nicknames and shortened versions of names, the only ones with changed names are:

Ulysses Simpson Grant, né Hiram Ulysses Grant
Gerald Rudolph Ford, né Leslie King Jr.
William Jefferson Clinton, né William Jefferson Blythe.

http://wiki.answers.com/Q/Why_did_Ulysses_S._Grant_change_his_name_from_Hiram_Ulysses_Grant_to_Ulysses_S._Grant
https://en.wikipedia.org/wiki/Gerald_ford
https://en.wikipedia.org/wiki/Bill_clinton

A possible fourth, Eisenhower is tricky because while he was at first named David Dwight, his mother reversed the names to Dwight David to avoid confusing him with his brother.
https://en.wikipedia.org/wiki/Dwight_D._Eisenhower#Early_life_and_education