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I have a zojirushi rice cooker. Usually I use it for japonica rice, occasionally for long grains, such as jasmine or basmati.

Firstly, you have to put in a minimum of one cup of rice in a zojirushi machine. Otherwise, there isn't enough steam generated to let it do it's job (EDIT - see below)

Rinsing may well be important. I never miss out this step as I understand how important it is with Japanese rice. Also, good rinsing will reduce arsenic levels by up to 50% (admittedly long thin grain rices typically have lower levels, but it's always good to reduce levels further).
I rinse using a classic Japanese technique.

Firstly - don't use the rice cooker bowl to rinse (it will reduce the life of the non stick coating considerably if this becomes a habit*).
Place the rice in a different bowl, add water and drain the rice. Once drained, firmly but carefully massage the rice between your hands. Do this at least 20 times. Add water and rinse the rice another three times or so until the water runs clear (you can collect this water to feed plants with). Work quickly so that the discarded white water isn't absorbed back into the rice - this can affect the taste.

Different kinds of rice, and differently aged rice from the same source, will require different amounts of water. Jasmine rice will need anything between 1 cup of rice to 1 or 1 and a quarter cups of water. I find that the indicator of how much water to rice on the zojirushi bowl can be a bit overgenerous.
When pouring in the water try and always do it when standing in the same place so that the angle of observation is consistent. Then check carefully to see if the water line is just under the mark, on it, or just above it. Results can change by quite a lot depending on where you are.

The amount IS very important though. Use 1 cup minimum to get the best out of your machine (ours is a small 3 cup model).

* There's actually a lot that will reduce the life of a rice machine bowl - as well as using it to wash and rinse rice, making brown rice, and particularly sautéing ingredients or adding oil to make mixed ingredient dishes will affect the bowl. We replaced an earlier Sanyo model with our current Zojirushi and took in a lot of the advice given to us from a Toshiba rice cooker expert (she was amazing) who we met at Yodobashi Camera in Asakusa. Two years later having endured heavy daily use and the bowl is still like new.

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Now I think about it, the settings in your zojirushi may have ¾ as a minimum amount because it's taking into account that 1 standard Japanese cup is less than an American one.
My kitchen has a few cups from Japanese 100yen stores so I always have one to hand when measuring for the rice cooker. My machine is for the UK/European market and the markings are like the ones in Japan.
In short, what is 1 rice cup for me may be marked ¾ rice cup for you.