I need to think about this. I'm not 100% on board for restricting image use with transparent background. If we do go with explicit white and dark backgrounds, we should also adjust the "dark" themes to use our colors.
It does depend on the design of the logo. The one for OpenWrt doesn't look good with grey filling. If we go with transparent background, we need a white (or dark, depending on the theme) buffer in and around the logo.
The definition of "background" is debatable. The square or rectangular icon is rarely elegant, yes, but having white areas isn't wrong, at all. Here are few examples: WhatsApp, Facebook, Google Translate, Outlook, LinkedIn, etc. In fact probably a very big part of logos include white areas and they don't set them as transparent.
Skype logo is a good example for using a white outline when the page background isn't white.