### Godot version

4.0.dev (61630d4e1e279278d29576b979b710b9025f473e)

### System information

Windows 10

### Issue description

> **Note**
> this issue was reported by @glimpsebeyond on [godot-vscode-plugin](https://github.com/godotengine/godot-vscode-plugin/issues/389#issuecomment-1302540744)

On Windows, due to a path separator mismatch (Godot: `/` and Windows: `\`), the debug adapter protocol fails as it thinks the opened project is in a different directory.

Likewise, there are other scenarios where this check is too strict, failing as well due to the drive letter's capitalization (`c:\` vs `C:\`. Good example from the linked issue:
```
[error] The editor and client are working on different paths; the client is on 
"d:\gameDev\gd4proj\builder\pawn.cs", but the editor is on "D:/gameDev/gd4proj/builder": CodeExpectedError: 
The editor and client are working on different paths; 
the client is on "d:\gameDev\gd4proj\builder\pawn.cs", but the editor is on "D:/gameDev/gd4proj/builder"
```

### Steps to reproduce

- Try using DAP from a project on Windows (like [VSCode's extension](https://github.com/godotengine/godot-vscode-plugin))

### Minimal reproduction project

N/A
#ifndef DEBUG_ADAPTER_PARSER_H
#define DEBUG_ADAPTER_PARSER_H
#include "core/config/project_settings.h"
#include "debug_adapter_protocol.h"
#include "debug_adapter_types.h"
struct DAPeer;
class DebugAdapterProtocol;
class DebugAdapterParser : public Object {
	GDCLASS(DebugAdapterParser, Object);
private:
	friend DebugAdapterProtocol;
	_FORCE_INLINE_ bool is_valid_path(const String &p_path) const {
		return p_path.begins_with(ProjectSettings::get_singleton()->get_resource_path());
	}
protected:
	static void _bind_methods();
	Dictionary prepare_base_event() const;
	Dictionary prepare_success_response(const Dictionary &p_params) const;
	Dictionary prepare_error_response(const Dictionary &p_params, DAP::ErrorType err_type, const Dictionary &variables = Dictionary()) const;
	Dictionary ev_stopped() const;
public:
	
	Dictionary req_initialize(const Dictionary &p_params) const;
	Dictionary req_launch(const Dictionary &p_params) const;
	Dictionary req_disconnect(const Dictionary &p_params) const;
	Dictionary req_attach(const Dictionary &p_params) const;