What is the first organic compound to be discovered ? Urea or alloxan?
The problem herein lies in the different possible definitions of organic chemistry.

The IUPAC defines an organic compound as a structure containing at least one carbon atom that can be named using the principles of organic nomenclature.

A less concise definition is

All compounds containing carbon and at least one other element that are not carbon monoxide, carbon dioxide or the latter’s inorganic salts.


Another one:

All compounds that contain at least one $\ce{C-H}$ bond.


Originally, the definition meant compounds isolated from organic materials i.e. organisms, living beings.


You see that the exact definition of organic compounds is difficult. But it was once very clear, historically: Anything not isolated from organisms or a derivative thereof was inorganic.
Note that isolation of organic compounds can be traced extremely far back in history: The ancient Egyptians isolated indigo as a colouring agent. However, they didn’t synthesise it and most attempts pre-19th century failed. Nobody questioned the possibility of isolating pure organic compounds from living matter, it was merely questioned whether they could be synthesised without involvement of organisms.
Alloxan was synthesised by oxidising uric acid with nitric acid as Wikipedia states. Thus, it was synthesised from an organic compound and would have been considered to contain enough vis vitalis even for further synthetic steps.
The compound your link states, 5-amino-3H-1,2,4-dithiazole-5-thione, is not described in any prime source as having been isolated from organisms. In fact, it is unknown to Wikipedia, so with all due skepticism, one can assume it to be of little to no relevance in living beings.
Compounds we today would call organic have been synthesised even earlier. Ethylene was first discovered in 1669 and it was successfully chlorinated to give 1,2-dichloroethane in 1794 — way before 1818, 1821 or 1828. Due to its name Dutch oil it is possible that it was considered an inorganic chemical. If it was synthesised from ethene that was, in turn, synthesised from ethanol, that would have just meant it carried over the vis vitalis from the original ethanol.
Wöhler’s discovery was that significant, because he used compounds that were (and are) unanimously declared inorganic (silver cyanate and ammonium chloride) and arrived directly at a compound that had previously been isolated from living material namely urea.