Eco paints are traditional natural paints such as lime wash or distemper

You could say that traditional natural paints are a subset of eco paints. After all they are ecologically kind. I think of eco paints as modern paints which utilize the most recent research and are always evolving. You could say that eco paints endeavour to overcome the problems that traditional natural paints have. Such as washability and coexisting with conventional chemical paints.

Eco paints are organic

The word organic means different things to different people. A chemist would say organic is anything that contains carbon, including oil. This would make conventional paints more organic than eco paints.
The use of the word organic for eco paints is just for marketing reasons, latching on to the organic food craze. Next they'll be using the word bespoke. Wait a minute isn't bespoke a little 2006!

Eco paints don't last for long

It's all about using the right product for the job. Linseed oil based paints can last 15 years on exterior wood before needing a recoat and silicate masonry paint will last 15 plus. This is because they kind of become the surface they are applied to rather than sitting on top waiting to peel.Some water based eco paints are labeled as being OK for exterior use but I wouldn't use them.

All eco paint ingredients are safe to human health and the environment

Any natural product in the wrong place can be damaging to the environment. I'm sure if you were to deposit tuns of carrot juice in a lake the fish wouldn't fare to well and the same with eco paints. We all know that there are many toxic plants.
Responsible eco paint manufacturers will choose non-toxic ingredients but remember anything in the wrong place can be a pollutant (in fact that's what a pollutant is. If I was to put flower petals in a tin of conventional chemical paint the petals would be the pollutant!). Lime is often used in eco paints and is caustic so can irritate the your skin. It's caustic powers also prevents mould growth, which is of course a good thing.

The use of eco paints is expensive

Yes, eco paints tend to cost a little more initially, about a third more but you save in other ways. Your less likely to need a primer, undercoat and topcoat system as the same product will often do all three. This can reduce wastage costs. Also many eco paints have better opacity meaning less coats. I would say the biggest cost saver is maintenance, as eco paints tend not to peel. This can reduce the redecorating costs by over half.