I have edited my post in order to please the displeased. I hope that, as it is, the right to expose my experience is now taken in. In an attempt to better understand why 'presidente' and 'presidenta' seem to be now widely accepted forms, it isn't enough by quoting RAE's present definition of these terms or their historical usage in certain parts of the Ibero-American territories, but it would require a deeper analysis of the social frame, historical background and contemporary challenges we, as a whole (individuals, organisations and governments), have been facing due to strong movements for equality of gender, feminism and the democratisation of culture (Internet is responsible for a great part of this). As other fellows here have stated, the term 'presidenta' may have been in usage for a long long time by some social groups and spanish-speaking places, however, one swallow doesn't make summer for everyone. Something has changed dramatically over the last decades: there has been global sociological changes, mass media intervention and access to cultural exchange on a mass scale, like never before. It can't be denied this reality has driven a lot of inconsistency in the usage of the Spanish language, over the last decades, in such a fast pace, that many, who wouldn't even qualify as elderly people, have been forced to accept (accepting doesn't been they think it is right) something that was embedded as wrong in their formative years. This is a tough exercise that unfortunately has become more frequent as years go by. I therefore would like to contribute to the explanation trying to provide a bring up some of that analysis, so the person who made this question doesn't just receive the 'RAE-says-so type of answer'. That simply doesn't explain why 'presidenta' is accepted as right nowadays. Let's have a different take:
I grew up in Santiago de Chile and Great Britain. I am a native Spanish speaker. Never in my childhood I was taught that neutral substantives could be masculine or feminine as well. They were either neutral or not, but not both, since something can't exist in two places at the same time. These inconsistencies were not practiced by well spoken people in Chile, and that has changed. Did this change due to media pressing for it? It is a clear representation of how, in the society we live in now, it would appear then to be OK to change ones mind about something, based on what is popular and not in what is common sense or consistent, which suggests we may be facing an era of absence of solid foundations and immutable principles, in which we change our minds for utilitarian reasons. More on this inconsistencies and mind flip-flopping can be read on the article from the New York Times 'The Mind of a Flip-Flopper' by Maggie Koerth-Baker on the following page www.nytimes.com/2012/08/19/magazine/the-mind-of-a-flip-flopper.html?_r=0
The term 'lideresa', for instance, I found it quite used and accepted during the time I lived in Mexico, but it was not something you could get away with in Chile without someone giving you an awkward look. We see then it may be appropriate, or not, depending on the cultural environment. We mustn't forget here that there has been, and there are, massive influences in the language spoken in some countries, coming from third countries. The perfect example here is Chile and Mexico/Spain. Many Chilean politicians were in exile in Mexico/Spain, got infused with Mexicanism/Spanish-isms and brought these back to Chile, naturally influencing the media and their social circle. You can read about this on the following source: 'Exiliados, emigrados y retornados. Chilenos en América y Europa, 1973-2004', del escritor José del Pozo Artigas ISBN: 956-284-498-6.
An even stronger influence takes place from the hand of the Mexican tv series which every day get to be watched by the average Chilean, the latter inevitably ending up transculturalised day after day. These are also Governmental decisions because, even when the Chilean administration choose not to, they could perfectly enforce the transmission of material mainly spoken or dubbed by Chilean-based artists, just like Spain enforces the broadcasting of most of the foreign films (even if these are of another Spanish speaking country) to be dubbed in Spanish, Iberic accent. As we see, political decisions have a lot to do with the way we end up transforming, manifesting and experiencing our language and culture all over the world. It is important to understand such decisions are not a collective work and as such, unfortunately, are taken only by a few. You can read further about the boom of the Mexican series in Chile in the following source from Univision: www.foro.univision.com/t5/Comunidad-de-Telenovelas/El-Boom-de-las-Telenovelas-Mexicanas-en-Chile-DA/td-p/211062995 . For further understanding of the degree in which individuals get influenced by TV language and in what degree these individuals integrate these influences in their own vocabulary, please refer to the following source, which lists further papers at bottom of page: html.rincondelvago.com/la-influencia-de-un-medio-de-comunicacion_la-television.html
The 'presidente de Argentina' (as 2015), herself, said she wanted to be addressed as 'presidenta'. Some did not agree, nevertheless, who are the general media to say no to that request? We know the result and proof is that most of the media ended up conceding the femenine form of presidente. Please do not confuse my statement; do not deem it a political view! I am attempting to illustrate the degree of power an administration has over the official and public usage of a language, in the end affecting us all. Just as rulers through history have even changed the religion of a country in order to achieve their own agendas. 'La presidente de Chile', Michele Bachelet, through permanent and oficial comunications, overriding the neutral/generic term stated in the Chilean constitution, which refers to 'Presidente', establishes her persona as 'La Presidenta de la República', which can be verified at the following source: 'Contacte con La presidenta' www.chileatiende.cl/servicios/ver/AA001'.
A good example of this are ministerial mandates. Keep in mind education of languages at school are directed by/through ministerial resolutions, a task of the government in place. You and I, learnt to use the language as a combination of social interaction and school studies and both imprint us forever. If the Ministry of Education wants you to refer to the 'president' as 'presidenta', include changes in the curriculum, modernisms, even remove certain previously accepted and taught matters, etc., you'll learn so at school, because teachers must comply with the government's educational resolutions. You can read further on how the educational curricula get periodically updated at the following sources provided, for instance, on the Ecuadorian Ministry of Education or the Argentinian Federal Council for Education.
If you disagree with this, it is perfectly fine, but don't make of your disagreement a motive to post false statements or trying to deny the undeniable. I am just providing true facts and not making a personal remark about how language is or should be learned, neither am I saying individuals in a society don't have free will or a personal choice at exercising their preferred way to speak. Government's educational curricula for language teaching, normally and in democracy, don't prevent society from self-determination when it comes to the act and ways of expression, however they do influence and you have read why.