Enforce schema validation on the publish and edit API level (#598)

<!-- Provide a brief summary of your changes -->

## Motivation and Context
<!-- Why is this change needed? What problem does it solve? -->
The following PR:
* Enforces schema validation on the API level (requests without schema
or older schema will fail)
* Updates the unit tests and examples to match that

## How Has This Been Tested?
<!-- Have you tested this in a real application? Which scenarios were
tested? -->

## Breaking Changes
<!-- Will users need to update their code or configurations? -->
**Note this can be considered breaking if clients have not included it
before.**

## Types of changes
<!-- What types of changes does your code introduce? Put an `x` in all
the boxes that apply: -->
- [ ] Bug fix (non-breaking change which fixes an issue)
- [ ] New feature (non-breaking change which adds functionality)
- [ ] Breaking change (fix or feature that would cause existing
functionality to change)
- [ ] Documentation update

## Checklist
<!-- Go over all the following points, and put an `x` in all the boxes
that apply. -->
- [ ] I have read the [MCP
Documentation](https://modelcontextprotocol.io)
- [ ] My code follows the repository's style guidelines
- [ ] New and existing tests pass locally
- [ ] I have added appropriate error handling
- [ ] I have added or updated documentation as needed

## Additional context
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---------

Signed-off-by: Radoslav Dimitrov <radoslav@stacklok.com>

docs/reference/server-json/generic-server-json.md

@@ -106,6 +106,7 @@ Suppose your MCP server application requires a `mcp start` CLI arguments to star
 
 ```json
 {
+  "$schema": "https://static.modelcontextprotocol.io/schemas/2025-09-29/server.schema.json",
   "name": "io.github.joelverhagen/knapcode-samplemcpserver",
   "description": "Sample NuGet MCP server for a random number and random weather",
   "version": "0.4.0-beta",
@@ -250,6 +251,7 @@ This will essentially instruct the MCP client to execute `dnx Knapcode.SampleMcp
 
 ```json
 {
+  "$schema": "https://static.modelcontextprotocol.io/schemas/2025-09-29/server.schema.json",
   "name": "io.modelcontextprotocol.anonymous/mcp-fs",
   "description": "Cloud-hosted MCP filesystem server",
   "repository": {
@@ -283,6 +285,7 @@ This will essentially instruct the MCP client to execute `dnx Knapcode.SampleMcp
 
 ```json
 {
+  "$schema": "https://static.modelcontextprotocol.io/schemas/2025-09-29/server.schema.json",
   "name": "io.github.example/weather-mcp",
   "description": "Python MCP server for weather data access",
   "repository": {
@@ -337,6 +340,7 @@ The `dnx` tool ships with the .NET 10 SDK, starting with Preview 6.
 
 ```json
 {
+  "$schema": "https://static.modelcontextprotocol.io/schemas/2025-09-29/server.schema.json",
   "name": "io.github.joelverhagen/knapcode-samplemcpserver",
   "description": "Sample NuGet MCP server for a random number and random weather",
   "repository": {
@@ -385,6 +389,7 @@ The `dnx` tool ships with the .NET 10 SDK, starting with Preview 6.
 
 ```json
 {
+  "$schema": "https://static.modelcontextprotocol.io/schemas/2025-09-29/server.schema.json",
   "name": "io.github.example/database-manager",
   "description": "MCP server for database operations with support for multiple database types",
   "repository": {
@@ -580,6 +585,7 @@ The `dnx` tool ships with the .NET 10 SDK, starting with Preview 6.
 
 ```json
 {
+  "$schema": "https://static.modelcontextprotocol.io/schemas/2025-09-29/server.schema.json",
   "name": "io.modelcontextprotocol/text-editor",
   "description": "MCP Bundle server for advanced text editing capabilities",
   "repository": {