As I understand rule 601, one should be able to cast a spell from their hand using mana from Lion's Eye Diamond, right? Or am I misinterpreting something there?

You are missing LED's errata, which says “Activate this ability only any
time you could cast an instant.”. That errata prevents you from using its
ability while anouncing spell, because you cannot cast instants during
(other) spell announcment - you can only activate mana abilities.

2013/1/2 Benjemin Harris <forum-2401@apps.magicjudges.org>

Lion's Eye Diamond's ability can be activated only any time you could cast an instant, meaning you need to hold priority.

Once you start casting a spell, you don't get priority until you've completely finished casting it. This means that you can't activate Lion's Eye Diamond in the middle of casting a spell; and you can't use its mana to cast the former.

Daniel Kitachewsky
L3, Paris, France
Rules NetRep

Okay. The errata is definitely what I missed. It threw me off because I thought it was still a mana ability. Thanks, guys!

It is a mana ability. It's just a mana ability with a restriction on when you're allowed to activate it.