Every class in 5E has built in subclass options.
The Basic Rules and Player's Handbook don't use the term subclass, but it's both obvious and helpful. Xanathar's Guide, an optional supplement, uses it as the chapter title for the section adding more of these significant, branching choices to each class. From that chapter, there's a nice clarification of the confusion you have:
Each class offers a character-defining choice at 1st, 2nd, or 3rd level that unlocks a series of special features, not available to the class as a whole. That choice is called a subclass. Each class has a collective term that describes its subclasses; in the fighter, for instance, the subclasses are called martial archetypes, and in the paladin, they’re sacred oaths.
The answer is: some classes have a thing called archetypes, but for the sorcerer, there is a thing called Sorcerous Origin. I guess technically you could tell your DM that you don't want this and forgo the benefits, but most of the interesting class features come from the origin.
For reference, the different subclasses are called:
\begin{array}{lll}
\text{Class} & \text{Subclass Name} & \text{Level Chosen} \\ \hline
\text{Barbarian} & \text{Primal Path} & 3^\text{rd} \\
\text{Bard} & \text{Bard College} & 3^\text{rd} \\
\text{Cleric} & \text{Divine Domain} & 1^\text{st} \\
\text{Druid} & \text{Druid Circle} & 2^\text{nd} \\
\text{Fighter} & \text{Martial Archetype} & 3^\text{rd} \\
\text{Monk} & \text{Monastic Tradition} & 3^\text{rd} \\
\text{Paladin} & \text{Sacred Oath} & 3^\text{rd} \\
\text{Ranger} & \text{Ranger Archetype} & 3^\text{rd} \\
\text{Rogue} & \text{Roguish Archetype} & 3^\text{rd} \\
\text{Sorcerer} & \text{Sorcerous Origin} & 1^\text{st} \\
\text{Warlock} & \text{Otherworldly Patron} & 1^\text{st} \\
\text{Wizard} & \text{Arcane Tradition} & 2^\text{nd} \\
\end{array}
So we can see that Fighter, Ranger, and Rogue, have "archetypes" — the rest have a path, a college, a circle, a domain, a tradition, or some other name.
Sorcerer subclass choices
For Sorcerer, from the free Basic Rules, your only Origin choice is
while the Player's Handbook adds
and Xanathar's adds
- Divine Soul
- Shadow Magic
- Storm Sorcery
As of right now, there are no other official subclass options for Sorcerer. For more variety, look into multiclassing, feats, or maybe simply choosing a backstory and really playing it up.
You will also find "homebrew" (non-official) origins of various quality and balance if you look online (it seems you found many of these), and there are also playtest Unearthed Arcana options — these are unofficial and unfinished (some of the are the test material for the ones that made it into Xanathar's). If you want to use one of these, work with your DM, and don't be surprised to hear "no", or "okay, but...".
Final notes
Finally, it's worth noting that many classes have other "path-type" choices which have some effect on the flavor of the class but aren't as significant as the subclasses (which often introduce whole new game mechanics not found elsewhere). For example, the ranger's Favored Enemy and the warlock's Pact Boon are class-feature options which have follow-on impact, but they're smaller in scope than a subclass.
In retrospect, it might have been nicer if the game designers had used the term "subclass" in the Player's Handbook to distinguish these very significant class branches. Instead, in supplementary material like Unearthed Arcana: Modifying Classes from 2015, the unfortunately-vague term "class option" is used, as in:
Each class contains at least one major decision point, referred to here as a class option.
(Strikethrough added because I don't want people skimming this answer to accidentally latch on to this outdated reference.)
I don't think archetype was ever used as the general term for these series of class features in anything official, but apparently some homebrew sites picked it up as a sort of jargon. It would probably be better and less confusing if everyone would shift to using subclass.
