What changes when you change format
Converting re-encodes the pixels; it does not improve them. Going from JPG to PNG keeps exactly what the JPG already holds — including its compression artefacts — in a lossless file that is usually several times larger. Going from PNG to JPG or WebP at a quality setting discards some detail to make the file smaller. Pick the output for where the image is going, not in the hope of higher quality.
Transparency
PNG and WebP store an alpha channel; JPG does not. When a transparent PNG, WebP, GIF or AVIF is converted to JPG, every transparent or partly transparent pixel is laid over the background colour you choose (white unless you change it). Converting a JPG to PNG cannot create transparency either: the JPG never had any, so the PNG has a solid background too. To cut a plain background out, use the transparent background maker on this site.
Why WebP is missing in Safari
The converter asks the browser to encode a one-pixel test image in each format and only offers the formats that come back correctly. Safari reads WebP but returns a PNG when asked to write one, so on an iPhone, iPad or Mac with Safari the outputs are JPG and PNG. AVIF is read by current browsers, but browsers generally cannot encode it from a web page, so AVIF is an input only here.
What is not carried over
Each image is drawn onto a canvas and encoded from there, so camera metadata (EXIF, including any GPS position), embedded colour profiles and animation are not kept. An animated GIF becomes a still image of its first frame.