_runtime_model_config persisted the live agent's RESOLVED provider into
the session row's model_config JSON. For any named providers:/
custom_providers: entry, agent.provider is the literal string "custom",
so the entry name was lost (and the api_key is deliberately never
persisted). On session.resume or _reset_session_agent the stored
provider="custom" fed resolve_runtime_provider(requested="custom"),
which cannot match a named entry — the rebuild either raised "No LLM
provider configured" or silently resolved placeholder credentials
against the patched-back base_url.

Persist the REQUESTED/entry identity instead: a new reverse lookup
find_custom_provider_identity(base_url) maps the endpoint URL back to
the canonical custom:<name> menu key. _runtime_model_config stores that
key; _make_agent performs the same recovery for rows persisted before
the fix, falling back to passing the stored base_url as
explicit_base_url so the direct-alias branch still targets the
session's endpoint when no entry matches.

Co-Authored-By: Claude Fable 5 <noreply@anthropic.com>

A thinking-only assistant turn (reasoning present, empty visible text) is
persisted with its reasoning fields and stays recallable from the transcript,
but `_history_to_messages` dropped it as "empty" before its reasoning was
attached. On desktop/TUI resume or reload the turn therefore vanished from the
session view while the agent could still recall it from a fresh session --
exactly the "messages disappear when the LLM uses its thinking block, but a new
session can recall them" symptom reported on #44022.

Keep an assistant turn when it carries reasoning, even with empty text, so the
desktop "Thinking…" disclosure has something to render. Genuinely empty turns
(no text, no reasoning, no tool calls) are still filtered out.

Refs #44022

Co-Authored-By: Claude Opus 4.8 <noreply@anthropic.com>