fix(insights-mcp): improve docs to clarify situation with libraries with MCP SDK and FastAPI (#15413)

## DESCRIBE YOUR PR

Contributes to
https://linear.app/getsentry/issue/TET-1328/fastmcp-documentation-onboarding

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- [ ] Checked Vercel preview for correctness, including links
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## EXTRA RESOURCES

- [Sentry Docs contributor guide](https://docs.sentry.io/contributing/)

Author: constantinius

docs/product/insights/ai/mcp/getting-started.mdx

@@ -51,12 +51,13 @@ const server = Sentry.wrapMcpServerWithSentry(new McpServer({
   minVersion="2.43.0"
 />
 
-The Sentry Python SDK supports MCP observability for [FastMCP](https://gofastmcp.com/getting-started/welcome). The integration automatically captures spans for your MCP server workflows including tool executions, resource access, and prompt handling.
+The Sentry Python SDK supports MCP observability for the [MCP Python SDK](https://github.com/modelcontextprotocol/python-sdk) (both low-level and FastMCP APIs) and [standalone FastMCP](https://gofastmcp.com/getting-started/welcome). The integration automatically captures spans for your MCP server workflows including tool executions, resource access, and prompt handling.
 
-#### Quick Start with FastMCP
+#### Quick Start
 
-```python
+```python {tabTitle:MCP Python SDK}
 import sentry_sdk
+from sentry_sdk.integrations.mcp import MCPIntegration
 from mcp.server.fastmcp import FastMCP
 
 # Sentry init needs to be above everything else
@@ -65,6 +66,7 @@ sentry_sdk.init(
     traces_sample_rate=1.0,
     # Optional: Enable to capture tool call arguments and results in Sentry, which may include PII
     send_default_pii=True,
+    integrations=[MCPIntegration()],
 )
 
 # Create the MCP server
@@ -79,3 +81,97 @@ async def calculate_sum(a: int, b: int) -> int:
 # Run the server
 mcp.run()
 ```
+
+```python {tabTitle:FastMCP}
+import sentry_sdk
+from sentry_sdk.integrations.mcp import MCPIntegration
+from fastmcp import FastMCP
+
+# Sentry init needs to be above everything else
+sentry_sdk.init(
+    dsn="___PUBLIC_DSN___",
+    traces_sample_rate=1.0,
+    # Optional: Enable to capture tool call arguments and results in Sentry, which may include PII
+    send_default_pii=True,
+    integrations=[MCPIntegration()],
+)
+
+# Create the MCP server
+mcp = FastMCP("Example MCP Server")
+
+# Define a tool
+@mcp.tool()
+async def calculate_sum(a: int, b: int) -> int:
+    """Add two numbers together."""
+    return a + b
+
+# Run the server
+mcp.run()
+```
+
+```python {tabTitle:MCP Low-Level API}
+import asyncio
+from typing import Any
+
+import sentry_sdk
+from sentry_sdk.integrations.mcp import MCPIntegration
+from mcp.server.lowlevel import Server
+from mcp.server import stdio
+from mcp.types import Tool, TextContent
+
+# Sentry init needs to be above everything else
+sentry_sdk.init(
+    dsn="___PUBLIC_DSN___",
+    traces_sample_rate=1.0,
+    # Optional: Enable to capture tool call arguments and results in Sentry, which may include PII
+    send_default_pii=True,
+    integrations=[MCPIntegration()],
+)
+
+# Create the MCP server
+server = Server("Example MCP Server")
+
+# Define tools
+@server.list_tools()
+async def list_tools() -> list[Tool]:
+    """List all available tools."""
+    return [
+        Tool(
+            name="calculate_sum",
+            description="Add two numbers together",
+            inputSchema={
+                "type": "object",
+                "properties": {
+                    "a": {"type": "number", "description": "First number"},
+                    "b": {"type": "number", "description": "Second number"},
+                },
+                "required": ["a", "b"],
+            },
+        )
+    ]
+
+# Handle tool execution
+@server.call_tool()
+async def call_tool(name: str, arguments: dict[str, Any]) -> list[TextContent]:
+    if name == "calculate_sum":
+        a = arguments.get("a", 0)
+        b = arguments.get("b", 0)
+        result = a + b
+        return [TextContent(type="text", text=f"The sum is {result}")]
+    
+    return [TextContent(type="text", text=f"Unknown tool: {name}")]
+
+# Run the server
+async def main():
+    """Run the MCP server using stdio transport."""
+    async with stdio.stdio_server() as (read_stream, write_stream):
+        await server.run(
+            read_stream,
+            write_stream,
+            server.create_initialization_options(),
+        )
+
+
+if __name__ == "__main__":
+    asyncio.run(main())
+```