fix: Use TypeScript private keyword instead of ES #private fields (#127)

* fix: Use TypeScript `private` keyword instead of ES `#private` fields in exported classes

ES private fields (`#field`) are included in `.d.ts` declarations and use
nominal identity, which causes type incompatibilities when multiple copies
of the package exist in a monorepo (e.g. pnpm installs). Switching to
TypeScript's `private` keyword keeps these fields out of the declarations
entirely, fixing the issue.

Fixes #126

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

* fix: Convert remaining #private fields in Connection class to private keyword

For consistency with the exported classes, also converts the internal
Connection class from ES #private fields/methods to TypeScript private.

Co-Authored-By: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

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Co-authored-by: Claude Opus 4.6 (1M context) <noreply@anthropic.com>

Author: chazcb