Arcane Magic and Different Mages Origins

Most common Scarterrans can tell the difference between a divine casters and arcane casters relatively easily.
Most divine casters, regardless of their affiliation or training cast their magic through their holy symbols and most divine casters figuratively or literally wear their piety on their sleeves.
Theurgists and favored souls may cast their divine magic near the same, most lay people understand that favored souls are born and theurgists are made.

To the common man, mages are very hard to tell apart. In most cases, the differences do not matter. Different mages may cast their spells in very different ways but the end results are the same.

Sorcerers

A sorcerer (male) or a sorceress (female) is born with innate ability to cast magic spells. In plural form, the term "sorcerers" is usually implied to be unisex. Sorcerers essentially gain their magical ability through accident of birth.
Nearly all dragons are sorcerers though not all dragons are skilled sorcerers. It is very common for mortals with draconic blood to have sorcerous talent manifest in their bloodline. Because they don't have a slew of other special abilities to develop in tandem with their sorcerer, distant dragon legacies are often better sorcerers than full dragons or first generation half-dragons.
Spirits can have children with mortals and mortals with spirit legacy occasionally manifest a talent for sorcery, usually but not exclusively if the spirits are affiliated with Greymoria.
Fair Folk often are sorcerers and they can have children with mortals, some mortals with Fae blood develop sorcery.
Children born and/or conceived near magic fonts and/or during magical surges occasionally grow up to be sorcerers. Occasionally, sorcerers created in this matter pass their arcane potential on to their children.
Swynfaredians preach that draconic sorcerers are superior to all other sorcerer types. In Kahdisteria, spirit legacies are considered the most favorable. Other subcultures may favor other sorcerous origins but they are rarely as outspoken.
To muddy the waters, a sorcerer and sorceresses frequently seek to marry and have children with each other and this means a sorcerer could have multiple magical legacies.
None of the causes of a person being born a sorcerer or sorceress work 100% of the time, it's an open question on whether the manifestation of sorcery depends on sheer dumb luck or if it's influenced by Greymoria or some other greater power. If Greymoria does have influence over who is a sorcerer or who is not, it is not known how or why the Mother of Magic chooses the recipients of her gifts or curses.

Wizards

A wizard (unisex term) is an ordinary person who learns to utilize arcane magic with specialized training.
The arcane laws of the universe are universal enough that anyone can theoretically use them. Much like how anyone with the right conditions can start a fire by rubbing sticks together, in theory anyone can cast a spell like a wizard.
Theory is not always practice. Otherwise everyone would be a wizard.
The core rule of magic is that magic has a price. It takes many years of highly focused study and training to be able to unlock even a simple spell though once a first spell is unlocked, the second and subsequent spells tend to come much quicker.
In order to commit to this training, a young person needs to devote a lot of time and energy to learning wizardry that they cannot spend learning to do much else. This creates an economic class barrier to people entering. Most lower class families need to have every family member pitching into the family trade, so most cannot afford to apprentice their sons and daughters to an older wizard, though this is not unheard of. Even high born people cannot always afford to have their children neglect the family business for a large chunk of their young lives.
Even if someone has the time to devote to learn wizardry, it takes a strong will, dedication and patience. Not everyone is willing to spend years and years studying magic with no tangible results for years on end and it takes a strong will to tug on the threads of the tapestry of Creation. A lot of would-be wizards burn out and quit before they learn their first cantrip or else their mentors terminate their training. Most wizards carefully screen potential apprentices before taking them on to try to avoid taking on a flunking candidate.
There are many different wizarding traditions which are outlined in the main wizardry article and satellite articles.

Warlocks and Witches

Warlocks (male) and witches (female) are individual who gained arcane power by making pacts with extraplanar beings or they are the direct descendants of warlocks and witches and inherited a pact with an otherworldly being that they didn't ask for.
The extraplanar beings are collectively called Patrons.  A Patron can be a spirit, Fair Folk, or even a Void demons.

Type

Metaphysical, Arcane