How should I understand the pronunciation of 驾驶证 (jiàshǐ zhèng) as 酱汁儿 (jiàng zhī er)?
Your 1st interpretation is correct. In a three-character word the middle character tends to be ommitted.

驾驶证     jia shi zheng -&gt;
驾驶证儿   jia shi zh-r  -&gt;
驾儿证儿   jia r zh-r    -&gt;
驾(儿)证儿 jia-r zh-r    -&gt;
酱汁儿     jiang zh-r


This is the same for 派出所 becoming 派儿所 (盼儿锁 in the article).