I doubt that they were named like that cause they ran around with lit grenades like lunatics. Is there some historic or strategic significance or is it just a name?
Side info: Currently playing a game called ,,Company of Heroes'' (WW2 strategy game) and it claims that Grenadiers were the core of the German army (time window of the game is the last two years of the war)
In 1943 Hitler renamed all units which until that time lacked a unit honorific as 'grenadiers.' While their is a historical significance to the name (early 17th-18th century black powder and pig iron grenades were heavy and temperamental, requiring brave, strong men to throw them) The significance of this was reduced after they were phased out during the early 1700s. Afterwards, despite the short-lived nature of these weapons, the name continued to carry these positive connotation. For example, British Army Infantry battalion deployed 'grenadier companies' on their right wing (the position of honour on the battlefield) and these continued to be filled with the largest and tallest men of the unit. Such physical discrimination or battlefield distinctions had altogether vanished when modern grenades had been reintroduced.
There was however a morale bosting connotation to the name, as it implied bravery, courage and physical ability. As I have also alluded to earlier, as Hitler had renamed much of his army 'grenadiers,' by dint of numbers they effectively became the core of the Heer.
Panzergrenadier-Divisionen were units with far higher numbers of vehicles compared to regular Infantriedivisionen this includes combat vehicles like Schützenpanzer and normal trucks. Panzergrenadierdivisions tried to have fully mobilized infantry battalions. They were often used more offensively since their increased mobility allowed them to be part of the spearhead and move with the tank units while infantry units were trailing behind securing flanks et cetera.
The distinction between normal infantry divisions and motorized division didn't exist in the Allied armies since all their divisions were fully mobilized even including a tank battalion which would include more tanks than many Panzerdivisionen fielded. The closest to a Panzergrenadierdivision would be the US infantry division. The US infantry division is far better equiped but still its the closest comparision. There is no counterpart to the normal German infantry division such unit would not be considered combat ready by Allies standards.
The term Grenadier lost its meaning at the latest with the introduction of Volksgrenadierdivsion.
I don't know if this sort of thing is allowed as a top-level reply, but I've never seen anyone use quotation marks like that. Where do you live, where that's the norm?