Custodial means a third party holds the keys and you hold a claim against them; non-custodial means you hold the keys and there is no counterparty.
Why it matters
This single distinction determines what happens when things go wrong. A custodial failure is an insolvency question answered by law; a non-custodial failure is final and has no appeal. See what self-custody actually means.
What you can check
The test is who can move the assets without your participation. Terms of service state whether client assets are pooled, whether the operator may lend or rehypothecate them, and what happens to them in an insolvency. A published on-chain address is not by itself evidence of segregation.
Where it showed up
Records in the Incident Tracker that turn on this: Mt. Gox collapse.
Not advice. Definitions are for understanding, not instruction. Nothing here is financial, legal, tax or security advice.