From what I understand, if a player has 4 artifacts in play, one of which is a lotus petal, he may announce Thoughtcast with affinity for 4, then sac the lotus petal to pay for the cost of the spell atÂ

.
Another relevant example that was particularly common in pre-Affinity-bannings Standard: if you have 4 artifacts in play, one of which is Chromatic Sphere, you can announce Thoughtcast, locking its cost at

, and then pay for it by filtering one mana through your Chromatic Sphere, netting you an extra card in the process,

.
Regarding Lion's Eye Diamond, yes, this trick doesn't work because Wizards errata'd it and added the "Play this ability only any time you could play an instant." text. Back in the day you used to be able to announce a spell, thus removing it from your hand, and then pay for it with Lion's Eye Diamond. I remember abusing this interaction in old Extended decks like PandeNought, where you'd want to pay for a Yawgmoth's Will that you had in hand with an LED, then replay it from the graveyard along with stuff like Pandemonium and Phyrexian Dreadnought, that would make for a very quick kill!
Luiggi