Split the single connection row of the gateway's shaped provider-error reply
into three: an interrupted established connection (reset/EOF/RemoteProtocolError),
a refused/unroutable endpoint (the case #86570 wrote the "not running or is
unreachable" wording for), and a cause-free SDK ``APIConnectionError`` that
supports neither diagnosis. A reset says nothing about whether the endpoint is
up, so telling the user to restart a server that just answered sends them to
debug the wrong thing (#116323).
Selectively adapted from Lei-k/hermes-agent#16 (497cc47556b1) via PR #109701,
rebased onto the current reply contract (rate-limit > auth > policy > connection,
every reply names a slash command, no operator jargon).
37c15650a7