Stepper motor - loss of steps

If it's broken then I assume it is the spiral cut type? I've had those break too but now I use the kind with the plastic spider. Just make sure the set screws are good and tight.

yes, the spiral ones.
I was thinking to switch to the ones with the plastic spider too.
I hope to find the type I need.
I got even new steppers (nema 23 again) but with bigger shaft (14 mm)
I should find the right couplers with my screw shafts (6 mm)

A useful search term is "Lovejoy Coupling" though that is a particular trademark for the generic " Jaw Coupling "

Though by the time you have bought all three parts, you can have spent a lot of money. Also, whilst the fact that you buy all the parts seprately means that you can mix shaft sizes, I am not finding it easy to find one that covers 14mm and 6mm in the same coupling size.
uk.rs-online.com/web/p/couplings/7019137/

coupeling shoudt be tight
so best is with a "little flex" TIMING belt
as well you can increase the ACC by 200% as it is more capable of Stepper microstep acceleration at least my expirience
and you can also get a better accuracy performence

Don't skimp on your coupling. Start by calculating the torque requirements of your motor and then compare that with the backlash free spec for the coupler. Make sure what you buy is capable of handling your torque. I started with a spiral one and ended up buying one from a motion control specialist locally. Its similar to a Lovejoy coupling. It cost me about $70 but at least I knew it was ft for purpose.

I'm finding it's not really easy to get the full coupler (6+spider+14).
One of the lot of reasons I love US, is the fact that you can find almost "everything", even if you're a "lesser being". Here in Italy if you're not a tycoon, it's really hard to find a company (don't dare to think about "shops") which will sell you something; and for sure, absolutely not cheap. And even if you find a "specialized shop" for "lesser beings" , the most of time they ask you "what the hell is that???".... specialized shops.....
Thus, I'll search the best solution I can, even the "chinese allied".
Change my setup from "direct driven" to "belt driven" won't be so easy (at least to me), so I think I'll keep my setup.