Should I add outlets inside my cabinets for LED lighting?
Your first plan was indeed no good...
As I and others pointed out in the comments, your initial plan to tap power for undercabinet lighting from a kitchen countertop receptacle was a blatant violation of NEC 210.52(B):

(B) Small Appliances.
(1) Receptacle Outlets Served. In the kitchen, pantry, breakfast room, dining room, or similar area of a dwelling unit, the
two or more 20-ampere small-appliance branch circuits
required by 210.11(C)(1) shall serve all wall and floor receptacle outlets covered by 210.52(A), all countertop outlets covered
by 210.52(C), and receptacle outlets for refrigeration equipment.
Exception No. 1: In addition to the required receptacles specified by
210.52, switched receptacles supplied from a general-purpose branch
circuit as defined in 210.70(A)(1), Exception No. 1, shall be permitted.
Exception No. 2: In addition to the required receptacles specified by
210.52, a receptacle outlet to serve a specific appliance shall be permitted to be supplied from an individual branch circuit rated 15 amperes
or greater.
(2) No Other Outlets. The two or more small-appliance
branch circuits specified in 210.52(B)(1) shall have no other
outlets.
Exception No. 1: A receptacle installed solely for the electrical supply to
and support of an electric clock in any of the rooms specified in
210.52(B)(1).
Exception No. 2: Receptacles installed to provide power for supplemental equipment and lighting on gas-fired ranges, ovens, or counter-mounted cooking units.
(3) Kitchen Receptacle Requirements. Receptacles installed
in a kitchen to serve countertop surfaces shall be supplied by
not fewer than two small-appliance branch circuits, either or
both of which shall also be permitted to supply receptacle
outlets in the same kitchen and in other rooms specified in
210.52(B)(1). Additional small-appliance branch circuits shall
be permitted to supply receptacle outlets in the kitchen and
other rooms specified in 210.52(B)(1). No small-appliance
branch circuit shall serve more than one kitchen.

Your revised plan will work though
The good news is that moving the undercabinet light outlet to a lighting branch circuit works just fine under the NEC, as there are no requirements for dedicated branch circuits to serve lighting in a dwelling unit, only receptacles.  My personal preference would be to use the kitchen lighting branch circuit to keep things logical for the next person who has to touch this, but there is no law that prohibits the kitchen undercabinet lights from going on the same branch circuit as the living room lights.