[AspireE2E] When dotnet watch on Aspire Starter App (ASP.NET Core/Blazor) app, The dotnet watch information displays incorrect after editing the file

**REGRESSION INFO:** Also reproes on Aspire 13.0.

**INSTALL STEPS**

1. Clean machine: Win11 x64 23h2 ENU
2. Install SDK 10.0.102
3. Install Aspire.ProjectTemplates 13.1.0

**REPRO STEPS**
1. Open CMD and create an Aspire Starter App (ASP.NET Core/Blazor)
```
dotnet new aspire-starter -o aspire01
cd aspire01/aspire01.AppHost
```
1. Run `dotnet watch`
1. After the dashboard load successfully, open webfrontend endpoint link on browser.
1. Open Home.razor file under xxx.Web, and add a `Counter` component, and save the file.

` `

**ACTUAL**
The `dotnet watch` say: The ApiService, AppHost and Web projects all Hot reload succeeded, but I just edit the Web project, it should just say only the web project Hot reload succeeded

**Note**:
1. This issue also reproduces when edit the file of ApiService project

**EXPECTED**
It should just say the web project Hot reload succeeded.