Convert JPEG Logo to Transparent

I already designed my logo into two separate styles and have it saved as a JPEG. I now need it rebuilt properly so the background is fully transparent and the edges remain crisp at any size. While the core design should stay the same, I’m open to a few subtle improvements—think light colour refinement or a slight polish to typography—nothing that changes the spirit of what I created.

Here is what I need from you:
• Redraw or vector-trace the current JPEG so it scales cleanly.
• Supply at least a high-resolution transparent PNG and an editable vector file (AI, SVG or EPS).
• Apply the minor tweaks we agree on and share a quick preview for approval before final export.

I will send the existing JPEG as soon as we start. If you have suggestions for small, tasteful enhancements, let’s discuss them upfront so we can wrap this up quickly.