What you found is for an LM465 Lamp Module. That uses Neutral and Line inputs to control a lamp on the output.
It does not apply to a WS467 that steals power through the Load and made for incandescent loads. This stealing power keeps LED bulbs doing things like pulsing, glowing all the time or make enough power line problems the WS467 stops responding to X10 commands.
I don't know if the modification to add a Neutral Power connection to the WS467 may make it work with a LED load.
It may or may not.
Also needs a Neutral power connection in the switch box. If you don't have a Neutral and only a switch loop. This would not work.
http://www.laureanno.com/Use the 'Quick Jump to X-10 Schematics' WS467 to see the modification drawn in.
You mentioned maybe changing to a module with a On Off relay in it. That will also require a Neutral connection in the switch box.
If the new under counter strip LED bulbs use a power supply to run them. Most will not work with a dimmer switch unless specifically rated OK on a dimmer switch.