The title of "Sith" isn't given by the public, it's more of a self-proclaimed title/group - at least after the fall of the original Sith Empire. One could argue that anybody claiming to be a Sith after Sidious is just kidding themselves and shouldn't be counted or from another point of view, anyone after the fall of the original Sith could be excluded - including everyone in the line of Bane. The only people with the right to exclude anyone would be "true" Sith - however we define that - and there's nobody left to speak up.
To answer your question, it seems all it takes to be able to call themselves "Sith" is identifying with some or most of the Sith ideals/philosophies and then labeling themselves as a new incarnation of the order. It doesn't really have anything to do with meeting a certain criteria.
Those philosophies include the Rule of Two (although this wasn't instituted until Darth Bane and was abandoned by Darth Krayt) and the Sith Code, authorized in 6900 BBY by Sorzus Syn and reinforced by Darth Bane, originally appearing in the Knights of the Old Republic video game:
Peace is a lie, there is only passion.
Through passion, I gain strength.
Through strength, I gain power.
Through power, I gain victory.
Through victory, my chains are broken.
The Force shall free me.
To illustrate my point, after Palpatine's death, the order lives on in 3 different cults, all of which were Sith more by self-proclamation than by classification:
(Excerpts taken from Wookieepedia)
The first to confront the new Jedi Order was Lumiya's Sith, a collection of darksiders loosely related to Darth Bane's Order. As an Emperor's Hand, Lumiya had a limited knowledge of Sith teachings and dark side techniques during her tutelage under Emperor Palpatine and Darth Vader. When both Sith Lords perished at the hands of each other at the Battle of Endor, Lumiya proclaimed herself as their successor; a Dark Lady of the Sith. A strict adherent to the teachings of Darth Bane, Lumiya preserved the Rule of Two in her Order, and thus she searched for only one Sith apprentice to serve at her side. After passing over two failed apprentices, Flint and Carnor Jax, the Dark Lady successfully trained Jacen Solo, a grandson of Darth Vader and a Jedi Knight of the Galactic Alliance, as her successor by 40 ABY. With Lumiya's guidance, Solo accepted the Sith mantle as Darth Caedus, the first active Dark Lord of the Sith since the time of Sidious and Vader.
Lost Tribe of Sith
In 43 ABY, the Jedi Order discovered the Lost Tribe of Sith, a hidden society that traced its origin to the Sith Empire of Naga Sadow. The Lost Tribe, created long before the rise of Darth Bane and the Rule of Two, was populated by many Sith individuals, all ruled by a Grand Lord. They formed a short-lived alliance with Luke Skywalker's Jedi Order to defeat a powerful being named Abeloth, but then betrayed the Jedi with preparations for galactic conquest.
The third organization that followed in the wake of Sidious's demise was the One Sith, founded by Darth Krayt. Once a Knight of the first Jedi Order, Krayt fought for the Old Republic during the Clone Wars as A'Sharad Hett. Although Hett survived the betrayal of the clone troopers during Order 66 and the subsequent Purge, his faith in the light side of the Force was destroyed by the desire for vengeance against the Galactic Empire and the Order of the Sith Lords. Hett eventually came to believe that the only way to defeat Sidious and Vader was by embracing the power of the dark side. Though he recreated himself as Darth Krayt, the self-proclaimed Sith Lord ignored Bane's Rule of Two. Under Krayt's Rule of One, the new Sith Order included many members in its ranks, all of whom were indoctrinated with absolute loyalty to a single Dark Lord of the Sith. In the year 130 ABY, Darth Krayt established a new Galactic Empire after seizing the Imperial throne for himself.