Different RFID tags and supported standards

Hello everyone,
While researching RFID standards, it seemed to me like there are mainly ISO 14443 and ISO 15693 standards that the manufacturers adhere to while specifying their tags.

I deduced that because of the modulation and power differences, a tag that corresponds to one of these standards will not work with the other. Am I right on this?
Also, I'm assuming readers that support one protocol will not support the other. Do correct me if I'm wrong.

I've been trying to figure standards that the main transponders in the market correspond to, like the Icode from Philips, Tagit from Texas Instruments, etc. but can't seem to find the information. And what the hell is Mifare standard?

Could anyone please shed some light?
Any help is appreciated.
Ananth.

Mifare :
https://www.nxp.com/#/pip/cb=[type=product,path=/53420/53422]|pip=[pfp=53422][0]

Fully compliant with ISO 14443A, MIFARE is the leading industry standard for contactless and dual interface smart card schemes. With an immense worldwide installed base, it is a proven RF communication technology for transmitting data between a card and a reader device.