I use Screen2Gif on Windows. However I use it to export to APNG, not to GIF (much better quality, which I keep as reference for later conversions). Then depending on limitations of every particular site I convert. For my web, I convert to WEBP animation (good compression compared to GIF/APNG, full 32-bit palette, supported in every browser except Safari (damn you Apple)). For itch.io, I have to convert to 3MB GIF, which is quite draconic limit IMO, but still achievable. For conversions I use FFmpeg, for further GIF optimization I use gifsicle tool. It took some time to master them as they're command line with lot of parameters, but in the end, it's much better than relying on online converters.
Hope someone will find it useful.