What are the key fields inside an X.509 certificate, and which three carry the core "identity + key + proof"?

The subject is the identity, the subjectPublicKeyInfo is the public key, and the signatureAlgorithm/signature at the bottom is the issuing CA's signature binding them together.

A typical X.509 v3 certificate contains:

• version (e.g. 3) and serialNumber — unique ID.
• issuer — the CA that issued it (e.g. O=Hochschule Luzern, CN=ISF-CA).
• validity — notBefore / notAfter dates (the cert expires!).
• subject — the identity the cert is about (e.g. CN=www.hslu.ch). ⇐ Identity
• subjectPublicKeyInfo — the algorithm + the public key itself. ⇐ Public key
• extensions — e.g. keyUsage: digitalSignature.
• signatureAlgorithm + signature value — the CA's signature over all the preceding fields. ⇐ Proof (CA signature)

Tip: Read a cert as a sentence: "The issuer swears that this subject owns this public key until this notAfter date — and here's my signature to prove it."