Maybe...
You sure it's not the ESP/Traction Control kicking in as it's happening in the wet.
More than likely, but then this would be accompanied by a tell tale flashing idiot light on the dash.
Worn out ditchfinders would produce the same effect as the traction control.
2 things to try; 1/ look for the idiot light when pulling away in first, and 2/ turn off the traction control (push and hold the button until you get a message that says something like "oh noes, you is gonna crash real bad if you isn't careful") and see if it still does it.
The OP has gone to the trouble of getting the car plugged in though, so I think he'd probably have noticed the idiot light flashing at him if he was just being lead footed. This kind of rules out traction control intervention...
There may also be another cause though, which I admit is pure speculaution on my part and would be considered outlandish if it weren't already common practice, and if VAG hadn't already proven themselves so adept at writing software that does funny things to engines. (did anyone else read the article about VW group cheating in the Sunday times last weekend? If I didn't already own one, I don't think I'd be buying one)
Just for a moment then, and because I've got the day off tomorrow and am well into a four can Stella sesh, let's hypothesise...
Given the sophistication and systems integration now built into these cars, and I think VAG are pretty much leading the field in terms of just how many bits of its cars talk to each other, it's not beyond the realms of possibility that some bright spark has decided to limit torque in first gear in certain situations. The use of multiple engine maps is now commonplace, with some manufacturers of certain bright red shouty Italian cars even going to the trouble of utilising a different engine map for each gear. I first came across a simpler version of this about 10 years ago in a Mazda 3 MPS though. The Mazda 3 lacked the AWD system of its larger stablemate, so it would deliberately limit torque in first gear to give the driver a fighting chance of getting off the line using only the front wheels whilst being equipped with an engine that developed rather more twistiness than the S3 of the day.
Could there a similar system on the A3 being driven by the OP? A rain and light sensor or even a simple "wipers on" signal could be used as a prompt to limit torque... Things like this have been done before for various reasons. Mazda used it to stop owners turning expensive tyres to smoke plumes, but my favourite was the Cadillac V8 in the '90s's that owners found ran like an absolute pig when the air con was turned off, stalling, stuttering and lacking in power. Weirdly though it was just fine when the air con was on. It turned out the emissions tests dictated that air con must be turned off, so Cadillac used the air con switch to to do something very VW, and change engine maps to something more environmentally friendly during official testing. Cadillac also echoed VW's efforts by getting caught...
How to test? If you have a rain sensor, cover it up so the car thinks it's not raining. If you have manu-matic wipers, turn them off. Now try pulling away in first and see if the problem is still there...
Can number 4 beckons though, so I'm not driving anywhere...