For IP-level communication, askubuntu.com/questions/22835/how-to-network-two-ubuntu-computers-using-ethernet-without-a-router/116680#116680 just worked between P51 and P14s both on Ubuntu 23.10 connected with a regular Cat 5e cable.
On both machines, first we found the Ethernet cable interface name with the ip CLI tool:
ip a
which outputs on the P41s:
1: lo: <LOOPBACK,UP,LOWER_UP> mtu 65536 qdisc noqueue state UNKNOWN group default qlen 1000
    link/loopback 00:00:00:00:00:00 brd 00:00:00:00:00:00
    inet 127.0.0.1/8 scope host lo
       valid_lft forever preferred_lft forever
    inet6 ::1/128 scope host 
       valid_lft forever preferred_lft forever
2: enp1s0f0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc pfifo_fast state UP group default qlen 1000
    link/ether fc:5c:ee:24:fb:b4 brd ff:ff:ff:ff:ff:ff
3: wlp2s0: <BROADCAST,MULTICAST,UP,LOWER_UP> mtu 1500 qdisc noqueue state UP group default qlen 1000
    link/ether 04:7b:cb:cc:1b:10 brd ff:ff:ff:ff:ff:ff
    inet 192.168.1.123/24 brd 192.168.1.255 scope global dynamic noprefixroute wlp2s0
       valid_lft 61284sec preferred_lft 61284sec
    inet6 fe80::3597:15d8:74ff:e112/64 scope link noprefixroute 
       valid_lft forever preferred_lft forever
so the interface was enp1s0f0, because wlp is wireless and lo is localhost.
So on the P14s we assign an IP of 10.0.0.10 to the P51:
sudo ip address add 10.0.0.10/24 dev enp1s0f0
Then on the P51 analogously, giving IP of 10.0.0.20 to the P14s:
sudo ip address add 10.0.0.20/24 dev enp0s31f6
And after that, P14s can:
ping 10.0.0.10
and P51 can:
ping 10.0.0.20
TODO after a few seconds, the settings appear to be forgotten, and ping stops working unless you do sudo ip address add on the local machine again. This seems to happen after a popup appears saying "Activation of network connection failed" as it fails to obtain Internet from the cable.
TODO list and delete such manual assignments we've made.