PNG files save with black background instead of transparent in Photoshop. Is there a solution?

I'm not really sure why this is happening. But I'm trying different image options for my website and wanted to see the difference between PNG and WebP for file size and transparency features. I can save with WebP normally and can have transparent backgrounds. However when I try to save as PNG, no matter what I do, the background always turns black. Even though I have no background set. Is this a missing feature or a bug? Has anyone else had this happen in Photoshop?

For reference, I uploaded the PNG image on the footer logo image on the bottom left side at the following address:

[link removed as per forum guidelines]

I wanted to use it on the entire website with a transparent background. But I stopped here until I hopefully fix it. If anyone has any tips or suggestions, please let me know!