Bolt-Era API Provider Widget Analysis

Lifted from `augment-it-bolt/specs/APIProviderWidget-Analysis.md`. Per-feature breakdown of the bolt monolith's API provider configuration widget — the surface that let users choose between Claude, GPT, and Perplexity per prompt section, with per-call options. The widget itself won't survive the rewrite verbatim, but the multi-provider configuration *shape* is the right starting point for the rewrite's model-routing UI.