Logo Vectorise & Shape Tweak

I have a raster copy of my logo that now needs to live everywhere—from tiny favicons to large-format print—so I need it rebuilt as a clean, fully-scalable vector. While redrawing, please adjust one specific shape element to refine the overall look; colours and text stay exactly as they are... (Budget: £10 - £20 GBP, Jobs: Adobe Creative Cloud, Adobe Illustrator, Adobe Photoshop, Graphic Design, Illustrator, Logo Design, Photoshop, Vector Design)