Haven’t heard back from @Wurkkos_Terry on this, but I’ve been using that same FC13 and USB-C-to-C cable, and for that same phone, and I noticed that the issue never happens as long as I started recharging the phone when it hits 10% or so – it only happens if I wait for the phone to discharge to 1 or 2%.
Not sure about what could be the root cause of that, but for now I consider the problem solved – just don’t wait until the phone’s battery is almost dead before starting to recharge it.