Short answer: it's not wrong, but it's linked to some complicated history and no clear distinction between the languages Dutch and German and their predecessors.
I'm no expert, but what I could quickly find about it:
It goes back to the late Middle ages when a early forms of Dutch and German were spoken: dietsc and duutsc refer to variants of the Germanic languages. These terms were mostly used to distinguish both Dutch and German from Roman languages and can apply to either Dutch or German. The distinction between Dutch and (low-)German (Nederlands en (Neder-)Duits) was not that clear.
The English word Dutch is derived from duutsc or duutsch
Also in Dutch the word Duits (current meaning: German) was used to refer to Dutch (Nederlands) up to the 17th century. This can for instance be seen in the Dutch national anthem (sung from the perspective of Willem van Oranje), the second line is:
ben ik, van Duitsen bloed (am I, of Dutch/German blood)
Also see this explanation in Dutch: https://nl.wikipedia.org/wiki/Nederlands_(naamsgeschiedenis)#.C3.9Eeudisk_in_de_volkstaal
All I'd really add to that would be that as I understand it the term(s) which ultimately became Dutch in English and Deutsch in German meant something like "vernacular" and/or "of our people". The Dutch used the words mentioned to designate themselves and their language as the Germans do today. We (English speakers) adopted their term as our term for them. However, we already had a term for the region that is now Germany (what was the the Holy Roman Empire), which was derived from the Latin Germania, so we used that instead to refer to Germany, German and Germans.
It's as OscarHermosilla mentioned, there's only one language spoken in The Netherlands, Belgium and Suriname. The only odd thing is that there is a term for the group of dialects spoken in Flanders, but there is no term for the group of dialects spoken in the Netherlands.
It's fine to say Ik ben een Vlaming en ik spreek Vlaams just as it's fine to say Ik ben een Brabander en ik spreek Brabants.
"there is no term for the group of dialects spoken in the Netherlands" Sure there is, at least in Belgium and most of the rest of the world, it's called 'Hollands'. Of course this is a historical inaccuracy caused by tradesmen from Holland spreading the concept of 'Holland' worldwide, but it is what it is.
It's best not to confuse the two: Flemish is not Belgian Dutch. Quick example sentences:
"Wat eet je graag?" "Nou, ik lust graag een patatje mét."
"Wat eet je graag?" "Nou, ik lust graag frietjes met mayonaise."
"Wa(d) ete geiren?" "Awel he, 'k mag 'kik geiren nekeêr e pakske frieten me mayonaise."
There's no standardised spelling and children are taught to read, write and speak the language of the Netherlands in schools, but Flemish does still get passed on orally through their parents, even if those parents think they're speaking proper Dutch to their child. The problem is that many people think their dialect/regiolect is completely unique and therefore there's no such thing as 'Flemish', but if you look past the differences in pronunciation of vowels, you'll see that most people speak basically the same way. The dialect continuum with the Netherlands, on the other hand, was severed long ago, and the difference is immediately apparent when you cross the border. There's a reason why our koterij stops right at the border.
(The 'eê' signifies a vowel distinction Dutch has already lost five hundred years ago, the soft-long vs sharp-long 'e': 'meer' is a lake, 'meêr' is more. There's also the soft-long vs sharp-long 'o', for that matter.)
By the way, Flemish for 'like a boss' would be 'gelijk nen baas'.
Okay, then the third pair of example sentences is Belgian Dutch, and considering Belgian Dutch is the official language of Flanders, a Flemish child that writes the third pair of example sentences in their Dutch class will get 10/10. If you call Flemish 'Belgian Dutch', you might as well call Norwegian 'Norwegian Swedish'.
You can't really trust Wikipedia articles based on information provided by an institute that has historically (and actually) tried to wipe out all non-Hollandish dialects from existence. They were successful in the Netherlands but never truly succeeded on the other side of the border, which is something that still angers the linguistic elite in Flanders.
Also, I'd just like to add that in Wallonia, people speak Belgian French, but also Walloon. Walloon is recognised as its own language, because there's no language union doing everything in its power to eradicate it. You can't just refer to Walloon by calling it 'Belgian French', because they're two completely different things. The same goes for Flemish and 'Belgian Dutch'.
Ik really don't agree. I give you one example. The different use in 'lopen' in the Netherlands en in Flanders.
It is not dialect when somebody uses 'lopen' when he/she means 'running'. It is used by all Dutch speaking people in Flanders. So why wouldn't I call it Flemisch?
Not like in German; scroll up and read all of the moderator Susande's responses. She has already answered this and here is another place to listen to different people saying this very sentence: https://forvo.com/word/ik_spreek_geen_nederlands/#nl
For the word spreken you could think of it as ik and spreek both ending in a k. However, it may be more useful to learn the general rules for Dutch verbs. Generally the 1st person singular form of a verb (that’s the ik form) doesn’t have any ending added to it (so spreek); all other singular forms add -t (so jij spreekt). Plurals add -(e)n and if there is a doubled letter in the middle it becomes a single one (so wij spreken). It isn’t always that simple but usually follows that structure. Important exceptions include hebben, which changes to jij hebt but hij/zij/het heeft, and zijn, which changes to ik ben, jij bent and hij/zij/het is.