Ticketed or not with their own seat, whether with a car seat or not in the cabin, the children under 2 years of age can typically sit on an adult caretaker's lap during take-off, landing and even turbulence.
In the case of some non-North American carriers, the lap-child is required to be fastened in with a lap-belt accessory attached to the fastened seat belt of the adult holding the child in the lap. In that case, the option is either to have one of those lap-belt attachments in use with the lap-child or to put the child in its own seat (with the seatbelt and/or in the car seat).
Originally Posted by
kavok
Data point, as someone who was in that exact situation 4 years ago:
The Delta FA wanted the infant in the car seat for take off and landing, even though the kid at that moment was “happier” in moms arms.
Could have been rogue FA. Could have been lesser known rule I was not aware of. Kid wasn’t as happy for takeoff and landing, but went to moms arms again to be comforted once we were up in the sky.
YMMV
There's definitely no "lesser known rule" that I've seen about this kind of thing on US, Canadian and European carriers for intra-jurisdictional (domestic/regional) flights or TATL flights. Out of a huge volume of flights where I've seen or known passengers to lap-carry a child during the flight but have a boarding pass for a separate seat or otherwise managed to get space for the infants to have their own seat, I've never seen anything more than a suggestion to put the child in the car seat for take-off/landing; and even that was rare and seemed to be a caring suggestion unless it appeared that the child was borderline 2 years old or otherwise too old to be a lap-child. Personally, my suggestion would be to put the child in the car seat as much as possible, especially if the car seat is already secured in place in the cabin for the child's use.