It's not about a tweet, just read your own quote.
You cannot authenticate with anything but Claude Code and Claude.ai.
But you do not need to authenticate with Claude Agent SDK (even though you can using env variables).
When you authenticate with Claude Code (allowed), Claude Agent SDK works without any further authentication.
It's really annoying that people keep trying to make this complicated because the inevitable end result is that they remove authless usage of the Agent SDK and save themselves the headache.
I really hate Clawdb-Moltb-OpenC-NanoCode or whatever half-baked project the grifters are on this week for ruining a good thing for the rest of us.
Did you read the second part? The ToS explicitly says that using the Agent SDK with Pro/Max authentication is disallowed. There is no ambiguity or interpretation here.