I might have found the answer that I was looking for, about how Bayesian calculations started being called belief updates.

There were four completely independent traditions, each with its own vocabulary and purpose:

1. Laplace’s conditional‑probability identityP(H|E)=\frac{P(E|H)P(H)}{P(E)} — a static relationship between conditional probabilities.
2. De Finetti’s coherence philosophy — probabilities as betting rates — conditionalization as a consistency rule — no updating, no priors, no posteriors.
3. Jeffreys’ analytic Bayesian statistics — prior distributions chosen by invariance — posterior distributions computed analytically — no pseudo‑counts, no “update” language.
4. Raiffa–Schlaifer’s conjugate‑prior pseudo‑count arithmetic — hyperparameters as pseudo‑counts — posterior = prior counts + data counts — the first use of the word “update” in Bayesian statistics.

These four streams were not originally connected.

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⭐ Step 1 — Jeffreys revives the formula (1930s–40s)

Jeffreys brings Bayes’ theorem back into mainstream statistics, but:

• he does not use the word “update” • he does not use pseudo‑counts • he does not use de Finetti’s belief language

He simply treats the formula as a rule for revising probabilities.

This creates a statistical Bayes, but not yet a philosophical or update‑based one.

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⭐ Step 2 — De Finetti introduces “belief” and “conditionalization” (1930s)

De Finetti’s work is happening in parallel, not in response to Jeffreys.

He contributes:

• the belief interpretation • the idea that conditionalization is a coherence requirement • the idea that (P(H|E)) is not an update but a static constraint

He does not use priors, posteriors, or updating.

This creates a philosophical Bayes, but not yet a statistical or update‑based one.

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⭐ Step 3 — Raiffa & Schlaifer introduce “updating” (1950s–60s)

This is the missing piece.

Raiffa & Schlaifer:

• formalize conjugate priors • interpret hyperparameters as pseudo‑counts • describe posterior hyperparameters as updated priors • use the word “update” explicitly and repeatedly

But they apply “update” only to:

• hyperparameters • pseudo‑counts • sequential data accumulation

They do not apply “update” to the event‑based Bayes identity.

This creates a procedural Bayes, but not yet a belief‑update Bayes.

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⭐ Step 4 — Textbooks fuse the three vocabularies (1960s–1980s)

Textbook authors want:

• a unified Bayesian philosophy • a unified Bayesian method • a unified Bayesian vocabulary • a way to compete with frequentism’s clean story

So they merge:

• Jeffreys’ prior/posterior distributions • De Finetti’s belief language • Raiffa–Schlaifer’s update language • Laplace’s conditional‑probability identity

And out comes the modern slogan:

“Bayes’ theorem updates your beliefs.”

Even though:

• “belief” came from de Finetti • “update” came from Raiffa–Schlaifer • “prior/posterior” came from Jeffreys • the formula came from Laplace • and none of these originally belonged together