Let's see. The minister of defense Cornelis de Witt was suddenly indicted for treason, and interogated under torture. The raadspensionaris (prime minister) was his brother Johan de Witt. He was also charged with treason, and locked up in the same prison, the Gevangenpoort in The Hague. The charges for both were at the time highly disputable. Interrogating under torture was no longer a normal procedure and highly questionable.
The mob bayed for their blood outside. The prison was under armed guard of the militia. By "sheer coincidence" the relief of the guard was incorrectly timed. The old guard had already left when the new guard hadn't arrived. The prison was unguarded. The mob grab both prisoners and horribly mutilated and killed them.
Nobody was convicted for negligence. It was just an unfortunate incident. Immediately after the brutal murder the Orange party - of course the prince had nothing to do with it, he terribly regretted the fate of the poor victims - purged and imprisoned most important members of the Staten party or removed them from office.
Yes, you could very well say this is a coup. It wasn't a coup because nobody called it a coup. One of many coups committed by the Orange party that weren't coups:
This wasn't a coup, but a dispute with the city of Amsterdam. Quit normal to move your army to lay siege in a dispute.
Did they come to power by legal means, or was there an armed overthrow of the federal government?
They came legally to power, there wasn't an armed overthrow of the government. Therefor, not a coup. In fact the political system at that time was a 2 party system: the Staten party and the Orange party. The Orange party was always present, but before this 'not a coup' as the opposition party.
Was there a purging of members of the States General or provincial governments?
Absolutely. The most famous person imprisoned was Hugo de Groot. He escaped his prison in a book chest (Dutch only). Loevenstein was used as the Dutch equivalent of Siberia for more important prisoners at that time.