proxy: Define service_info metric showing the run state (#12749)

## Problem

Monitoring dashboards show aggregates of all proxy instances, including
terminating ones. This can skew the results or make graphs less
readable. Also, alerts must be tuned to ignore certain signals from
terminating proxies.

## Summary of changes

Add a `service_info` metric currently with one label, `state`, showing
if an instance is in state `init`, `running`, or `terminating`. The
metric can be joined with other metrics to filter the presented time
series.

proxy/src/binary/proxy.rs

@@ -40,7 +40,7 @@ use crate::config::{
 };
 use crate::context::parquet::ParquetUploadArgs;
 use crate::http::health_server::AppMetrics;
-use crate::metrics::Metrics;
+use crate::metrics::{Metrics, ServiceInfo};
 use crate::rate_limiter::{EndpointRateLimiter, RateBucketInfo, WakeComputeRateLimiter};
 use crate::redis::connection_with_credentials_provider::ConnectionWithCredentialsProvider;
 use crate::redis::kv_ops::RedisKVClient;
@@ -590,6 +590,11 @@ pub async fn run() -> anyhow::Result<()> {
         }
     }
 
+    Metrics::get()
+        .service
+        .info
+        .set_label(ServiceInfo::running());
+
     let maintenance = loop {
         // get one complete task
         match futures::future::select(