On the difference between mage, wizard and sorcerer:
Mage:
TES II description - "Mages are those that devote their lives to the study and practice of the arcane magical arts. An experienced mage can be a deadly force to reckon with";
TES III description - "Most mages claim to study magic for its intellectual rewards, but they also often profit from its practical applications. Varying widely in temperament and motivation, mages share but one thing in common - an avid love of spellcasting.";
TES IV description - "Prefering to use their extensive knowledge of all things magical, they wield a might more powerful than the sharpest blade.";
Legends description - "Mages harness Intelligence and Willpower to unleash the power and fury of Magnus";

Sorcerer:
TES II description - "Sorcerers are quite adept at the manipulation of magic, although they do not generate their own magical energy. Rather, they absorb the energy of spells cast at them, and use this energy to power their own spells";
TES III description - "Though spellcasters by vocation, sorcerers rely most on summonings and enchantments. They are greedy for magic scrolls, rings, armor and weapons, and commanding undead and Daedric servants gratifies their egos.";
TES IV description - "Besting the most well-equipped fighters, they rely on the spells of the mystic arts. Unique to these mages is the bodily stamina to be armed with the thickest armor.";
Legends description - "Not afraid of a staff or a mace, Sorcerers' Intelligence and Endurance make them formidable foes.";
ESO Sorcerer description - "Sorcerers can use conjuration and destruction spells to hurl lightning bolts and create shock fields, wield dark magic to snare and stun, and summon Daedric combat followers from Oblivion to assist them".

On the wizards. First, here is my definition a character's class: a character's class is a predefined complex of skills and knowledge, a result of a particular course of study which grants a character the ability to perform certain functions within the social division of labour in any system so that a character may specialize (specialization). In other words, game class is a type of education a character acquires. For example, characters with a Sorcerer education can have many professions: a healer, a tank, an enchanter, a soldier, a scholar, etc. It's only the matter of certain skills they use, their views, beliefs and.. some economic demands - the benefits of some educational programs may be changed from patch to a patch.
Wizard is not a class, but rather used in the game as a general term to name all the magicka users whatever class they are, e.g. Telvanni wizards. Mage, on the other hand being a class, is also used as a synonym to Wizard and is used that way too.

Are mages in ESO born with magic? According to the Imperial Geographical Society:
"Imperial Theosophy teaches us that our world was born from magicka, the creative force that informs and sustains all life."
"Whereas Oblivion may surround us every night, it is aetherial energy that infuses our daily existence, from highest to lowest, and gives all the races of men, mer, and beast common purpose. Its magic brings the rain to the fields, love to our hearths, and scientific principles to our technological industries".

Oh, yes, I've forgotten about the ranks.. Thank you, @VaranisArano!