Reduce distributor code duplication

[zhuoyuan-liu]

What this PR does / why we need it:
This PR reuses the existing function updateMetrics to reduce code duplication.

Checklist

• Reviewed the CONTRIBUTING.md guide (required)
• Documentation added
• Tests updated
• CHANGELOG.md updated
• Changes that require user attention or interaction to upgrade are documented in docs/sources/upgrading/_index.md

[CLAassistant]

All committers have signed the CLA.

[grafanabot]

./tools/diff_coverage.sh ../loki-target-branch/test_results.txt test_results.txt ingester,distributor,querier,querier/queryrange,iter,storage,chunkenc,logql,loki

Change in test coverage per package. Green indicates 0 or positive change, red indicates that test coverage for a package fell.

+           ingester	0%
+        distributor	0%
+            querier	0%
+ querier/queryrange	0%
+               iter	0%
+            storage	0%
+           chunkenc	0%
+              logql	0%
+               loki	0%

[grafanabot]

./tools/diff_coverage.sh ../loki-target-branch/test_results.txt test_results.txt ingester,distributor,querier,querier/queryrange,iter,storage,chunkenc,logql,loki

Change in test coverage per package. Green indicates 0 or positive change, red indicates that test coverage for a package fell.

+           ingester	0%
+        distributor	0%
+            querier	0%
+ querier/queryrange	0%
+               iter	0%
+            storage	0%
+           chunkenc	0%
+              logql	0%
+               loki	0%

[MichelHollands]

LGTM. @zhuoyuan-liu Thanks for this cleanup!