[release] Core stable release 1.13.0 updates (#6552)

Co-authored-by: opentelemetrybot <107717825+opentelemetrybot@users.noreply.github.com>

Author: Kielek

Directory.Packages.props

@@ -2,7 +2,7 @@
 
   <PropertyGroup>
     <ManagePackageVersionsCentrally>true</ManagePackageVersionsCentrally>
-    <OTelLatestStableVer>1.12.0</OTelLatestStableVer>
+    <OTelLatestStableVer>1.13.0</OTelLatestStableVer>
   </PropertyGroup>
 
   <!--

src/OpenTelemetry.Exporter.Prometheus.AspNetCore/CHANGELOG.md

@@ -7,6 +7,9 @@ Notes](../../RELEASENOTES.md).
 
 ## Unreleased
 
+* Updated OpenTelemetry core component version(s) to `1.13.0`.
+  ([#6552](https://github.com/open-telemetry/opentelemetry-dotnet/pull/6552))
+
 ## 1.12.0-beta.1
 
 Released 2025-May-06

src/OpenTelemetry.Exporter.Prometheus.HttpListener/CHANGELOG.md

@@ -7,6 +7,9 @@ Notes](../../RELEASENOTES.md).
 
 ## Unreleased
 
+* Updated OpenTelemetry core component version(s) to `1.13.0`.
+  ([#6552](https://github.com/open-telemetry/opentelemetry-dotnet/pull/6552))
+
 ## 1.12.0-beta.1
 
 Released 2025-May-06

src/OpenTelemetry.Shims.OpenTracing/CHANGELOG.md

@@ -6,6 +6,9 @@ Notes](../../RELEASENOTES.md).
 
 ## Unreleased
 
+* Updated OpenTelemetry core component version(s) to `1.13.0`.
+  ([#6552](https://github.com/open-telemetry/opentelemetry-dotnet/pull/6552))
+
 ## 1.12.0-beta.1
 
 Released 2025-May-06