fix: Use TypeScript private keyword instead of ES #private fields

[chazcb]

Summary

• Replaces ES #private fields with TypeScript's private keyword in all classes: AgentSideConnection, ClientSideConnection, TerminalHandle, and the internal Connection class
• ES #private fields appear in .d.ts declarations and use nominal identity, causing type mismatches when multiple copies of the package exist (common in pnpm monorepos)
• TypeScript's private keyword keeps these fields out of the declarations entirely, fixing the issue
• Note: This trades runtime encapsulation for compile-time only. The fields become regular JS properties at runtime, but they are internal implementation details not intended for consumer use.

Fixes #126

Test plan

• tsc builds successfully with no type errors
• All 94 tests pass
• .d.ts output no longer contains #private for exported classes

🤖 Generated with Claude Code

[benbrandt]

I think we only need to change the private members and not the methods if I understand the symptom correctly?

These are the only # in the .d.ts?

export declare class AgentSideConnection {
    #private;

export declare class TerminalHandle {
    #private;
 
export declare class ClientSideConnection implements Agent {
    #private;

declare class Connection {
    #private;

[chazcb]

I think we only need to change the private members and not the methods if I understand the symptom correctly?

These are the only # in the .d.ts?

export declare class AgentSideConnection {
    #private;

export declare class TerminalHandle {
    #private;
 
export declare class ClientSideConnection implements Agent {
    #private;

declare class Connection {
    #private;

I was thinking we should go for consistency throughout, but agree we can just change the public interface. If you feel strongly about it I'll back out the changes to the internals