Alright. My bass player buddy keeps sending me postings from here.
Having written a few thousand charts (that people have paid for), and learned a great deal about the craft from my pops, here's my two cents, which is a basically summation of a few things said before:
Ami triad = A C E
Now it was vague whether or not the original poster was saying whether or not the 5th is still included in the chord. So here goes:
Fma7/A = A C E F, though most keyboard players by default are going voice the right hand either F A C E, or E F A C.
One thing my pops said, and I agree--If you want it played a certain way, tell them. An easy, no brainer way to do that:
Am(add F). Problem solved. 9 times out of 10, it'll be voiced as an Am chord with an added F on top (A C E F)
If the 5th is NOT supposed to be played:
F/A will suffice, but do note that colorwise, unless it's recognized and played as part of a progression (below) that provides more insight into the voicing, keyboard players may omit the 3rd in the right hand to provide more consonance (so, like a F5/A or Fomit3/A), the David Foster thing (which I'm guilty of doing on poppy ballads) is to read an F/A as an Fsus2/A (omit the 3rd and play the 2nd instead). So if an F/A is out by itself and you're concerned how it might be voiced, F/A is pretty concise theory-wise, but it can be colored MANY ways depending on the player, the style, and the context.
If you're talking about a progression standpoint (as mentioned before: Am Am+5 Am6 Am+5) then:
I'd much rather see:
Am Am+5 Am6 Am+5 (the Scherer books do this)
than
Am F/A Am6 F/A
In theory they mean the same thing, but guitar players in particular are less likely to jump around to grab an F-based voicing).
In fact, it's getting to be more common to see:
Am +5 6 +5
Which clues the player in further that it's a progression and not a massive change.
I guarantee that if you start bringing b13 b6 notation into the mix, you will confuse a portion of players. I've seen a good number of charts I have written where b13's and colors have been CROSSED OUT, to make things easy for a lazy player. If I know I'm writing for a non-jazz crowd (i.e. a bunch of top 40 players with questionable reading chops), I'll cut my losses and simplify changes (i.e. C7b5 instead of C7#11) in the interest of the material having a chance at keeping its coloration.
FWIW, a few bassists I play with (great players, not great readers) will go through charts ahead of time and cross out the part of the chord change that's not their root note.
Knowing how something is most likely to be read makes a huge difference. There's always a fine line between factual/theoretical correctness, and giving the player the right information to reproduce your musical intent, which 9 times out of 10 is the purpose of sheet music. Debate theory all day long--it's healthy--but if it makes it more difficult to get your point across to the player, always go with the most simple solution =)