what I don't understand is if it's the plug, then it would cut in and out when you plugged it into the outlet directly, not just when you tried going through the timer. changing a plug (polarised or grounded) is very simple.
can you post a picture of the plug ?
That's not always true, and would be more inline with the wires in the cord meeting the plug. But this sounds like the actual plugs ends. Odd manufacturing variances might mean the wall outlet just had more surface area to get a connection than the timer(s), and/or the plug from the cord has shorter leads that just cannot reach the internals of the timer, or the timers (both the same model) were sourced with parts that had less contact area that is deeper inside.
The fact it works when wigged on both timers is good news.
$3 fix with a new generic plug, and is almost a certain fix after reading all these posts.
The post about trying an extension cord to the wall and then to the timer would prove this. If that doesn't work with the timer, then try the extension cord with one of the lights that did work. Then if that Doesn't work, you know the outlets on the timer are just weird, and you just open them up and put in a new outlet there....or even just use a plier to bend the metal tabs outwards.