What a fantastic collection!
For a go-to check, I ran the logos from a meme that was running on Twitter for some time ("Nothing is original: these are not the logos of Medium, Airbnb, Flipboard, and Beats": https://twitter.com/stefanotirloni/status/725065684862922753...) -- and behold, all the old ones pop up but none of the new ones.
How can you hold the copyright on designs which are trademarks? I'm quite confused by "All rights reserved" in the footer. So, I must ask: are you trying to create an open directory or is this a proprietary project?
You can claim copyright on your rendition (all black versus the original color; or the text creating the logo in CSS or SVG) of the trademark in question. Whether that would hold up in court is another matter entirely.
That said, the site owner certainly holds the copyright on the site design, layout, textual content, etc. Just because it's a directory/portfolio of trademarked logos drawn by the artist doesn't mean the other content isn't covered.
One of the oldest and the largest logo references is Logo Lounge, which you might've seen in a printed form as a set of smaller square books, each chokeful of logos. Been around for 10-15 years minimumum, massively influential and well-respected.
Unlike other sites it costs $100 to become a LogoLounge member for a year and be able to add logos to their collection. Works well to filter out lower quality amateurish work that is prevalent on other sites.
A friend of mine does logo work for small businesses. They almost always ask for logos that aren't really logos. In short, things that wouldn't be recognizable when rendered in just black and white.
This might be helpful to get that message across with customers. I especially like the navigation via designer, country, style, etc.
When doing branding work I often distinguish between symbols as basic shapes and logos combining those shapes with brand name and/or slogan. In my personal taxonomy this is a library of symbols as opposed to logos.
Didn't take long to notice that Azuma Drive In and Airbnb share the same logo. Old news. Probably impossible to determine if this is plagiarism or merely coincidence.
Perhaps the source is this book? Logo Modernism: https://www.amazon.co.uk/Logo-Modernism-Jens-Müller/dp/38365... (source: still the Twitter thread.)