The 8KB vs. 6,466 bytes difference is due to the block size on your disk. Block size are typically 4 or 8KB, and each file takes up an increment of the block size. But when you transmit the file via the web, only 6,466 bytes of bandwidth will be used.
Part of the 5.513K vs. 6,466 bytes difference is because a KB is 1024 bytes, not 1000. So your file is 6,466 bytes = 6.314KB
The remaining 0.8KB is a mystery to me. Some could be metadata. Some could be because Photoshop estimates what the size will be and is not always right. Now I'm curious about it too.