Originally, the word "presidenta" didn't even exist. Presidents in history were always men and the word was only used in a masculine form. Languages are constantly changing, and now that there are female presidents, "presidenta" is a common and generally accepted word.
Considering that "she" is female, the word "presidente" would have to be female too. One way to do this is to change the "el" in "el presidente" to "la." However, now that people commonly use "presidenta," "la presidenta" is also an acceptable way to do it.
Generally, when you change the gender of the noun it will do one of the two or the other. (It will change either the article, "el" to "la," or it will change the article and the word to a feminine form) However, considering that in terms of language "presidenta" is a newer word, either way is still considered acceptable.