[docs] Updates for StartWithHost removal

docs/trace/customizing-the-sdk/README.md

@@ -34,13 +34,12 @@ using OpenTelemetry.Trace;
 var appBuilder = WebApplication.CreateBuilder(args);
 
 appBuilder.Services.AddOpenTelemetry()
-    .WithTracing(builder => builder.AddConsoleExporter())
-    .StartWithHost();
+    .WithTracing(builder => builder.AddConsoleExporter());
 ```
 
 > **Note**
 > The
-[StartWithHost](../../../src/OpenTelemetry.Extensions.Hosting/README.md#extension-method-reference)
+[AddOpenTelemetry](../../../src/OpenTelemetry.Extensions.Hosting/README.md#extension-method-reference)
 extension automatically starts and stops the `TracerProvider` with the host.
 
 ### Using Sdk.CreateTracerProviderBuilder
@@ -442,16 +441,15 @@ appBuilder.Services.AddSingleton<MyCustomService>();
 
 appBuilder.Services.AddOpenTelemetry()
     .WithTracing(builder => builder
-        .AddProcessor<MyCustomProcessor>())
-    .StartWithHost();
+        .AddProcessor<MyCustomProcessor>());
 ```
 
 When using the `AddOpenTelemetry` & `WithTracing` extension methods the
 `TracerProvider` does not own its `IServiceCollection` and instead registers
 into an existing collection (typically the collection used is the one managed by
 the application host). The `TracerProviderBuilder` will be able to access all
 services registered into that collection. For lifecycle management, the
-`StartWithHost` registers an [IHostedService
+`AddOpenTelemetry` registers an [IHostedService
 ](https://learn.microsoft.com/dotnet/api/microsoft.extensions.hosting.ihostedservice)
 which is used to automatically start the `TracerProvider` when the host starts
 and the host will automatically shutdown and dispose the `TracerProvider` when
@@ -651,8 +649,7 @@ appBuilder.Services.Configure<JaegerExporterOptions>(
     appBuilder.Configuration.GetSection("OpenTelemetry:Jaeger"));
 
 appBuilder.Services.AddOpenTelemetry()
-    .WithTracing(builder => builder.AddJaegerExporter())
-    .StartWithHost();
+    .WithTracing(builder => builder.AddJaegerExporter());
 ```
 
 The OpenTelemetry .NET SDK supports running multiple `TracerProvider`s inside
@@ -696,6 +693,5 @@ appBuilder.Services.Configure<JaegerExporterOptions>(
 appBuilder.Services.AddOpenTelemetry()
     .WithTracing(builder => builder
         .AddJaegerExporter(name: "JaegerPrimary", configure: null)
-        .AddJaegerExporter(name: "JaegerSecondary", configure: null))
-    .StartWithHost();
+        .AddJaegerExporter(name: "JaegerSecondary", configure: null));
 ```

docs/trace/extending-the-sdk/README.md

@@ -635,7 +635,7 @@ this case).
 > **Note**
 > `ConfigureOpenTelemetryTracerProvider` and
 `ConfigureOpenTelemetryMeterProvider` do not automatically start OpenTelemetry.
-The host is responsible for either calling `StartWithHost` in the
+The host is responsible for either calling `AddOpenTelemetry` in the
 [OpenTelemetry.Extensions.Hosting](../../../src/OpenTelemetry.Extensions.Hosting/README.md)
 package, calling `Build` when using the `Sdk.CreateTracerProviderBuilder` and
 `Sdk.CreateMeterProviderBuilder` methods, or by accessing the `TracerProvider`

docs/trace/getting-started-aspnetcore/README.md

@@ -78,12 +78,11 @@ appBuilder.Services.AddOpenTelemetry()
         .AddService(serviceName: "OTel.NET Getting Started"))
     .WithTracing(builder => builder
         .AddAspNetCoreInstrumentation()
-        .AddConsoleExporter())
-    .StartWithHost();
+        .AddConsoleExporter());
 ```
 
 > **Note**
-> The `StartWithHost` extension is part of the
+> The `AddOpenTelemetry` extension is part of the
 [OpenTelemetry.Extensions.Hosting](../../../src/OpenTelemetry.Extensions.Hosting/README.md)
 package.
 

src/OpenTelemetry.Api.ProviderBuilderExtensions/Metrics/OpenTelemetryDependencyInjectionMetricsServiceCollectionExtensions.cs

@@ -39,7 +39,7 @@ public static class OpenTelemetryDependencyInjectionMetricsServiceCollectionExte
     /// <item>A <see cref="MeterProvider"/> will not be created automatically
     /// using this method. To begin collecting metrics use the
     /// <c>IServiceCollection.AddOpenTelemetry</c> extension in the
-    /// <c>OpenTelemetry</c> package.</item>
+    /// <c>OpenTelemetry.Extensions.Hosting</c> package.</item>
     /// </list>
     /// </remarks>
     /// <param name="services">The <see cref="IServiceCollection" /> to add

src/OpenTelemetry.Api.ProviderBuilderExtensions/Trace/OpenTelemetryDependencyInjectionTracingServiceCollectionExtensions.cs

@@ -39,7 +39,7 @@ public static class OpenTelemetryDependencyInjectionTracingServiceCollectionExte
     /// <item>A <see cref="TracerProvider"/> will not be created automatically
     /// using this method. To begin collecting metrics use the
     /// <c>IServiceCollection.AddOpenTelemetry</c> extension in the
-    /// <c>OpenTelemetry</c> package.</item>
+    /// <c>OpenTelemetry.Extensions.Hosting</c> package.</item>
     /// </list>
     /// </remarks>
     /// <param name="services">The <see cref="IServiceCollection" /> to add