If the gateway returns two providers that resolve to the same display name
(e.g. `kimi-coding` and `kimi-coding-cn` both → "Kimi For Coding"), the
picker now appends the slug so users can tell them apart, in both the
provider list and the selected-provider header. No-op when names are
already unique.
Refs #10526 — the Python backend dedupe from #10599 skips one alias, but
user-defined providers, canonical overlays, and future regressions can
still surface as indistinguishable rows in the picker. This is a
client-side safety net on top of that.
