As Brian stated, you may have a marginal signal at that outlet, and the RR501 is picking it up better than the AM486. You stated that the AM486 works in other applications - was it using the same HouseCode/Unit code that you tried it with for your landscape lights? If it is a marginal signal (and/or noise interference), trying a different housecode/unitcode combination might make a difference.
One additional suggestion: Using an AM486 or an RR501 outdoors might not be the best idea - they aren't rated for that.
Black and Decker used to make some outdoor-rated modules (they came as a short grounded extension cord with the module as an in-line block), but I haven't seen those in a few years.
In my house, I decided to extend the landscape lighting transformer's AC line cord into the house, and plug it into the Appliance Module on an outlet in my basement.