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I dont think the timing is the problem otherwise it would happen at all temperatures. You could replace the 7x crankshaft position sensor just to be safe incase the sensor is beginning to fail. After replacement do the following procedure after replacement. Let me know if you have questions, thanks. Let me know if you have more questions, thanks.
From what you said about the key trick this backs up the problem as likely being the ignition switch if it was going bad it was getting really sensitive to the position of the key in the ignition. With that being said I would start by replacing the ignition switch. Here is a picture location of the crank sensor. Let me know if you have questions. Thanks
2003 Jetta left rear window blue plastic part of the