The cheapest RFID/NFC "card" tag is an expired tap-to-use transit ticket. It has a unique ID that can be used to trigger an iOS Shortcut to take any action, e.g. speak an audio description, open URL, open app, etc.
In a simple sense, NTAG cards can do NFC things, but MIFARE can do lots more (access control for example)..and also NFC things..somewhat.
Magic mifare refers to special cards that let you bypass the write-lock of genuine mifare cards. These are mostly used for cloning keys (either for red-team pentesting or for people who want a copy of an office key for whatever reason)
MIFARE Classic uses a proprietary and mandatory encryption/authentication algorithm and is therefore not ISO 14443-4 compliant. As a result, NFC-compliant readers don't need to support it, and in fact non-NXP ones (including many popular Android phones) usually don't.
On the other hand, as you say, MIFARE Classic supports capabilities beyond NFC/NDEF, but there are fully NFC-compliant tags that do so as well (e.g. MIFARE DESfire, which properly stacks encryption in an ISO 14443-4 compliant way).
Many of my cards are just repurposed from other things. Lots of hotel keycards can be rewritten to open URLs on phones for example.
I actually have a hotel keycard taped to my washing machine to do some laundry-based automation with my phone. Maybe I should write about that sometime..
If you're getting an implant already, why not make it an actual smartcard that you can use for WebAuthN, GPG, SSH etc.? :)
On the other hand, the fear of permanently bricking it or messing up the GlobalPlatform card management key has so far prevented me from doing it myself...
Because those cost $350 as opposed to $89, and the install only costs $60, and there is no stopping you from implanting more than one in different locations.
Many people get the small xEM or xM1 first to play with.
If you only care about static data that you can optionally freeze to a permanent read-only mode, NTAG21[3|5|6] tags are probably among the cheapest ones that reliably work with all iOS and most Android devices.
MIFARE Classic supports (completely broken and for this use case) useless encryption and doesn't work with some Android devices, as they're not really a part of the standardized NFC stack.
If you want to get really fancy, you can also get a Java Card based smartcard and install an NDEF application yourself. You could then also install a FIDO application and use the same card as a "homebrew Yubikey" :)
For purchase, there are many form factors: https://store.gototags.com/nfc-tags
On Amazon, search NTAG215.