do i need a new engine starter?

I finally had my customer's 1997 Maxima overnight in the 2 degree weather ...

(Ultrastart 1270TL has been working flawlessly until the frigid temps have set in .. anything say .. below 25 degrees, tach successfully learned at a nice, normal idle.. tach source is a good low voltage AC source from ECM)

I hit the STAR key .. here's what happens:  the horn chirps once, and (this is strange) .. The actual CAR STARTER starts spinning .. but IS NOT ENGAGED with the engine!! The starter spins so fast in about 0.5 seconds that it stops spinning (r/s thinks it started the car) and the parking lights come on and STAY on (to indicate that the r/s successfully started my car).

When I hit the STAR key for two seconds after this, it shuts the r/s down (two chirps, park lights go out). 

When I now hit the STAR key a third time, it chirps once, and the starter this time ENGAGES successfully with the engine, it gets the engine started NO PROBLEM. 

So... now I wait 6 hours... and go out the customer's vehicle (in the cold) with just the little old key .. i get into the freezing cold car, put the key in, turn it and LOW AND BEHOLD .. the first attempt at starting the car with the customer key .. the STARTER ITSELF just spins like crazy (this high pitched whiny spinning sound .. like a power drill) then after about 1.5 seconds seems to UNSTICK and engages into the engine (with a nice little chatter for a split second) and finally starts the engine.

<Sorry for the long post, but hopefully this will help someone, hopefully me!>

So, do I need to suggest to the customer to get a new starter? .. and actual Nissan engine starter? Is the cold just making it "stick" initially?  Because in temps warmer than 25 .. there is NEVER a problem either with the key start or the remote start.

Thanks.

Thanks again guys, here's what I did:

I verified that the OE starter#1 wire was in tact and produced a good 12+ voltage when key is in the crank position

I also verified that starer#2 wire was in tact, as well as my separate relay connected to it (I brought a separate power line from the battery into the cabin to power my second starter wire: 85 = 2nd starter optional output from Ultrastart 1270TL, 86 = chassis ground, 87 = power line from battery 12v+, 30 = second starter wire)

Sounds like it's a bad starter from your guys' advice .. gunna suggest he take it to Nissan -- he's just glad we figured this out.  Thanks again for all your help.