Codex cloud is now called Codex web. It looks like OpenAI's Codex cloud (the cloud version of their Codex coding agent) was quietly rebranded to Codex web at some point in the last few days.
Here's a screenshot of the Internet Archive copy from 18th December (the capture on the 28th maintains that Codex cloud title but did not fully load CSS for me):

And here's that same page today with the updated product name:

Anthropic's equivalent product has the incredibly clumsy name Claude Code on the web, which I shorten to "Claude Code for web" but even then bugs me because I mostly interact with it via Anthropic's native mobile app.
I was hoping to see Claude Code for web rebrand to Claude Code Cloud - I did not expect OpenAI to rebrand in the opposite direction!
Update: Clarification from OpenAI Codex engineering lead Thibault Sottiaux:
Just aligning the documentation with how folks refer to it. I personally differentiate between cloud tasks and codex web. With cloud tasks running on our hosted runtime (includes code review, github, slack, linear, ...) and codex web being the web app.
I asked what they called Codex in the iPhone app and he said:
Codex iOS
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