Your understanding of what they are is closer to reality, than what you've read in the wiki.
BIOS is a firmware, old standard, for x86 PCs.
EFI and UEFI are just different names for the same firmware - first it was called EFI and then got renamed into UEFI, so the former refers to the older (initial versions) of it, the latter for everything afterwards. UEFI is a firmware and it's not tight to x86 at all. It is more advanced and easier to use than BIOS, certainly. You need to write a loader for your OS in C, that makes heavy use of numerous services provided by UEFI. They are indeed very numerous - taking into account all the protocols, that might be included.
As of firmware examples. apart from the aforementioned BIOS and UEFI, one might remember OpenFirmware (OpenBoot), ARC stuff, totally disappeared now I guess. This is from real - fully fledged firmwares, with a strong and well defined interface, made for using it when loading an OS. There is also one firmware from the moronic set,
absolutely incapable, crippled down thing, that makes me wanna destroy it.
It's called uboot. such a poop. no interfaces, instead - brain damaging following linux, like this is the only OS on the world. Well, you got the idea. it's crap.
There is also firmwares for a particular controller/device - WiFi chips, AHCI controllers, internal SD/eMMC/SSD controllers firmwares etc. they are kinda "local" firmwares, whereas BIOS, UEFI, OF are global, platform firmwares.
Finally, if you wanna know about UEFI, go and download the official specification. First sections are the best and pretty easy way to get the idea of what it is and what it's not.