Creates a provider credential for the active account. In an organization, Braintrust and migrated model providers (currently OpenAI, Anthropic, Google, xAI, Mixlayer, Vercel AI Gateway, Tinfoil, Amazon Bedrock, Vertex AI, Vertex AI (Claude), and generic-openai) create organization-owned connections and return connection IDs. For OpenAI, Anthropic, Google, and xAI, isDefault=true selects organization BYOK authority; false stores the connection without changing custody. Mixlayer, Vercel AI Gateway, Tinfoil, Amazon Bedrock, Vertex AI, Vertex AI (Claude), and generic-openai creates are always storage-only; use the organization-policy endpoint to select one explicitly. Personal accounts and providers that have not migrated retain provider-key ownership. Requires BYOK entitlement.