If I understood your question correctly, your mistake is here:
reac = 2000; RT.push(reac); $("#RT").val(RT);
You push and show reaction time as soon as you show word to type.
Key is to first show the word and then start counting and not do it concurrently.
By the way there is still a little problem with reseting t1 in wrong place, so if you hit the keypress before fadeOut, but after default reaction time has fired, you'll get result like "2200".
You can fix it by, for example, putting it out of fadeOut: fiddle
To make it also change every time you press a key you can modify it like that: fiddle