Is a document really safer here than in a cloud editor?

Safer in one specific, checkable way: it is not on somebody else's disk. A cloud editor's privacy is a policy, an access-control implementation and a breach history you cannot inspect. Here the document is in your browser's own storage or a file on your disk, and you can verify that nothing left by watching the network tab while you work. What it is NOT is backed up — see the trade-offs.

What happens to .docx formatting in each?

A cloud editor imports a .docx into its own internal model, which is usually simpler than Word's, and exports an approximation on the way out. This editor keeps Word's model — named styles, numbering definitions, section properties, fields — and counts anything it cannot represent rather than dropping it silently. Both lose something; only one tells you what.

Can several people edit at once?

Yes, but by a different mechanism. A cloud editor's server is the shared copy everyone connects to, which is robust and works between strangers. Here a session is peer-to-peer between browsers, arranged by a link whose keys sit in the URL fragment so the signalling server never sees them. That is stronger on privacy and weaker on reach: everyone has to be online at the same time.

What do I lose by not having an account?

Sync, sharing by e-mail address, permissions per person, and recovery when your device dies. Those are genuine features of an account, not artefacts of one, and there is no account here to provide them. Exporting a .docx and putting it wherever you already keep files is the substitute, and for many people it is what they were doing anyway.

In-Browser Word Editor vs Cloud Word Processors

Both run in a browser tab. Only one keeps the document on your machine — what that changes about privacy, offline use, .docx fidelity and collaboration.