If you want to get technical, it's flash ROM chips; more akin to an SD card than SSD (though the technology between the two isn't far apart these days). But for all intents and purposes Nintendo's Game Cards are cartridges. They're designed in the spririt of cartridges and thusly are often referred to as cartridges.
But they're not cartridges, regardless of how people refer to them. To be a cartridge, as traditionally applied to video games, the ROM must be directly accessible from CPU space (whether completely, or through banking).
They might be designed in the spirit of cartridges, but they load files into RAM from a filesystem, and never access them directly from the storage media. Thus they're fancy SD cards that really, really want to be carts, but aren't.
Cartridges died with the GBA. Unless you count the myriad unlicensed, bootleg, or knockoff consoles that exist with multigame carts.
Edit (30 minutes later): The inherent nature of cartridges also allows direct access to peripheral chips (coprocessors, etc) found in the cartridge in CPU space as well.
You're confusing typical implementation as a technical definition. There's no actual rule which states a "cartridge" has to follow that definition (and in fact some 8bit micro computers with support for cartridges didn't follow your specification).
Given modern systems have a fat OS rather than a thin layer of firmware like the consoles of old, it would but highly illogical to build a cartridge system to the identical specifications of the 70s to mid 90's consoles. But that doesn't diminish the literal definition of the term "cartridge" just because you happen to nitpick the technical implementation.
If we're going to quibble about the definition of cartridge, I'd submit a far more interesting dimension is the ability of a cartridge to extend the console in meaningful ways, in the way that a SNES can not, on its own, run Star Fox.
That new definition covers off the majority of 8 bit micros (to the best of my knowledge) but I believe the CPC 464 Plus (read: not the regular CPC 464 but a later gaming model) couldn't be extended by it's cartridges; and frankly why would you need to as it had its own expansion port in addition to the cartridge slot.
I'd also like to add that I wasn't quibbling over the definition of "cartridge". I understand the term is quite broad but I am happy with that recognised definition even if it isn't a technical description. I know us engineers have a need to describe technology but not everything needs to be jargon.
I was just trying to inject an interesting angle to the discussion, not trying to be precise about what they are. It was an interesting characteristic they had for a while there. Personally I'm satisfied with "if it looks like a cartridge, it is", in which case absolutely yes the Switch uses cartridges.
