Logo Vectorization & Engraved Favicons

My current logo is available only as a high-resolution PNG, and I need it rebuilt as a clean, fully-scalable vector in Adobe Illustrator. The design features a clothing button; once the vector is complete, I’ll supply two initials so you can add them to the button and make them appear genuinely engraved (subtle depth, shadows, and highlights rather than a flat overlay).

When the engraving looks right, please export the button by itself and transform it into:
• A complete favicon set (.ico plus 16 × 16, 32 × 32, 48 × 48, and 180 × 180 PNGs)
• Square social-media profile images (at least 800 × 800 PNG and SVG)

Acceptance criteria
• Vector lines are crisp, neatly organised in layers, and delivered as an .ai file with a PDF-compatible preview
• Engraving effect feels embossed/engraved at 100 % zoom with no pixelation on export
• All favicon sizes render sharply on dark and light backgrounds
• Colour, proportions, and spacing match the original PNG exactly unless we agree on tweaks

Please keep the working paths editable and include any type you use as outlined shapes so there are no font conflicts. I’ll be quick with feedback and the initials once you send the first vector draft.