I have googled the Internet and got only instructions for old RouterOs versions.

I have recently bought a MikroTik router. I have installed the basic options with Quick Set.

Now I want to bind MAC addresses to static IPs, just as I had in previous two routers.

I tried to WebFig/ARP/Add New. However, nothing happens. The IP of the device is still the same, even if I reset the device.

What I am doing wrong?

You should do this in menu IP-->DHCP, not ARP

IP->DHCP Server->Leases

Easiest way is to let device get dynamic address and then select Make Static, so you won't have to type MAC address yourself. You can then change IP address if you want and device will get it next time.

I tried that too. What happens that after a while I have two addresses for one MAC on the leases list, and the one that is working is the one I don't want!

You'll have to wait a bit longer, until original lease (which you don't want) expires. DHCP server can not push another address to the device, device has to go into re-new procedure during which server assigns new address.
How long do you have to wait? Until timer in column "Expires After" doesn't count down to 0. But that's the latest time. Most of devices start re-lease procedure at half-time.

Are you by any chance doing something wrong with Client ID field? Like using it as comment or something? It would break things. That's the only explanation that I can currently think of, how you can get two entries for same MAC address.

You can change it, it's why the field is there, the problem is that it's something different than you think. :) I'll admit that I don't know too much about it myself, for example where it's actively used. It's some identifier chosen by client and server can use it as another condition besides MAC address. And it's exactly what happened, you added something, client sent something else, condition didn't match, so client got another address.

If you just need to describe the entry, you can use comment, WebFig has it as separate field at the bottom, WinBox has it behind another button.

If you're looking for hostnames you can use on other devices, there's currently no good solution in RouterOS. It's possible to use lease script to add static DNS entries. But unless you have multiple local subnets, it's probably best to ignore it and let devices use mDNS (automatic hostnames under .local TLD).