Please update the game to show what is said with /say above your char's head as well as in the chat log.
Other chat types (/tell, /shout, etc.) should remain in chat log (to reduce clutter as well); only possibly with the exception that /tell-ing another player should somehow notify them of the tell.

It is my honest and humble opinion that currently (imo devs have done a great job of cleaning things up) the single biggest issue plaguing this game is the lack of in-game communication.
I have seen many posts on various forums about why that is, but not one has hit the nail on the head.

The biggest issue with FFXIV's social development is the fact that you can NOT see other players trying to talk to you unless you happen to be looking at your chat box at that exact moment. And not everyone thinks of running around you using /wave emotes until they get your attention, nor wants to.

Nobody wants to bother chatting with people when there is a 90% chance they will not get a reply!!

(I spend hours in game sometimes wishing other ppl would talk to me only to find out they tried, and I missed it because I was busy at a behest or something else. Nobody wants to sit and watch their chat box for 10 minutes every time they message someone to see if they get a reply)