### 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
if (!DebugAdapterProtocol::get_singleton()->get_current_peer()->attached) {
		EditorRunBar::get_singleton()->stop_playing();
	}
	return prepare_success_response(p_params);
}
Dictionary DebugAdapterParser::req_launch(const Dictionary &p_params) const {
	Dictionary args = p_params["arguments"];
	if (args.has("project") && !is_valid_path(args["project"])) {
		Dictionary variables;
		variables["clientPath"] = args["project"];
		variables["editorPath"] = ProjectSettings::get_singleton()->get_resource_path();
		return prepare_error_response(p_params, DAP::ErrorType::WRONG_PATH, variables);
	}
	if (args.has("godot/custom_data")) {
		DebugAdapterProtocol::get_singleton()->get_current_peer()->supportsCustomData = args["godot/custom_data"];
	}
	ScriptEditorDebugger *dbg = EditorDebuggerNode::get_singleton()->get_default_debugger();
	if ((bool)args["noDebug"] != dbg->is_skip_breakpoints()) {
		dbg->debug_skip_breakpoints();
	}
	String platform_string = args.get("platform", "host");
	if (platform_string == "host") {
		EditorRunBar::get_singleton()->play_main_scene();
	} else {
		int device = args.get("device", -1);
		int idx = -1;
		if (platform_string == "android") {
			for (int i = 0; i < EditorExport::get_singleton()->get_export_platform_count(); i++) {
				if (EditorExport::get_singleton()->get_export_platform(i)->get_name() == "Android") {