it seems that routers decapsulate the frame on arrival, and
encapsulate the packet in a frame in order to send it.
Yes. A router must strip off the layer-2 frame in order to get to the layer-3 packet. The router then routes the packet to the next interface toward the destination, based on the layer-3 destination address. At the next interface, it must build a new frame for the packet for the layer-2 protocol on the next interface, which could be a completely different layer-2 protocol than the one used on the first interface.
But why people talk about packets forwarding ?
That is what the router is doing. It forwards packets from one interface to another interface (one network to another network).
Also, when you ping a domain name, you have "packets transmitted,
received". Is it an abuse of language?
No. Ping is an application that uses ICMP echo requests and replies, and ICMP is an integral part of IP, which is a layer-3 protocol that uses packets.
So, if I eavesdrop the link between two routers, do I see frames (with
mac addresses) or packets ?
Not all layer-2 protocols use MAC addresses, Only the IEEE LAN protocols use MAC addresses. Often, the layer-2 connection between routers will use something like PPP, which has no addressing on the layer-2 frames because there are only two possible endpoints. If the connection between the two routers is an IEEE protocol, then, yes, you will see frames with MAC addresses. If it is frame relay, you will see frames with DLCIs; with ATM, you will see frames with VPI/VCI; etc.