There are many different flavors of the Markdown specification for writing documents. They all agree on the basic elements, so for really straightforward Markdown documents - where each Markdown element is used in isolation from other elements - any Markdown rendering software will produce the same results.
Things start to become more complicated when you start using extensions of the core specification, like tables or strikethrough fonts. A further complication is when you begin using the basic elements in combination, like using header elements within block quotes, or even using paragraphs within list items.
Stata (as of version 15) has two different Markdown specifications built into it's markdown command (and therefore, also into dyndoc and dyntext). The default specification is flexmark, and you also have the option (currently undocumented) to use pegdown. These differ slightly from each other, meaning the same Markdown document might come out looking different from each other. They also differ slightly from the pandoc default specification, which is used in German Rodriguez's markstat command or when working in Rstudio.
(Technical aside: both flexmark and pandoc are configurable to emulate other Markdown parser/renderers. While the behavior of pandoc can be changed through the use of command line options, changing the flexmark emulation would require you to dig into the Java code.)
As a writer, you will probably want to pick one specification and stick with that. However, a downside to all three of these specifications is that they are not particularly well documented in the details. In what follows, I try to fill in some of those details.
Although most descriptions of Markdown syntax point to Gruber`s original specification, two of the most thoroughly specified and described flavors of Markdown are commonmark and gfm (Github-flavored Markdown, which aims to be a strict extension of commonmark). Especially because flexmark aims/claims to be a commonmark emulation by default, it is useful to work through these specifications, in order to reveal where they agree and/or disagree.
These examples are rendered with flexmark, using Stata's markdown command.