Like everyone else said, it depends on what you're looking for: Christmas decorations and spirit, or lower crowds. I would NOT say that the first week of January yields LOW crowds. They will definitely be MUCH lower on your January dates than the December ones, but if it's anything like last year...the crowds were not "low". We went on the first Monday following the holidays when school started (I think it was January 4 or 5). I did NOT expect it to be so crowded. I figured everyone would be back to school, work, etc. That was not the case. We sat in line in the parking garage for 15 minutes just to buy our parking ticket. Waited another 35 minutes to buy park tickets. Took 15 minutes of standing in line to get into the park. Wait times were not horrible though. I think Space Mountain was about 45-60 minutes or so. Big Thunder was about 40-50. Dumbo and Peter Pan were both about 40. A lot of the Christmas decorations were down and Haunted Mansion and IASW were both closed. We spent about 4 hours at California Adventure and it was less crowded than DL. I had DD and we went to the Bug's Life area and rode those rides over and over again with no wait. DS and DH went to the big rides and I think they rode 4 rides in about 4 hours. It wasn't horrible and the crowds were definitely manageable, and like I said...the January crowds will be MUCH lower than December. I just don't think that "low crowds" is the right term for that time and can be misleading. I wouldn't expect to walk onto anything and I would expect to be dealing with lots and lots of people. I was SHOCKED at how crowded it was when we went!! I really expected "low crowds", but that wasn't the case. I think low-ER is more accurate
Anyway, I think it comes down to this: December = Christmas decorations and festivities, special Christmas overlays (HM and IASW), long hours (I think 8:00 - 12:00?), HIGH crowds. January = little Christmas decorations, no special festivities, HM and IASW closed, shorter hours, lowER crowds. I think it's really a question of whether you want to be there for the holiday festivities and Christmas spirit or not. If not, I would go in January
I just want to add also that everyone's experiences are different, so my experience in no way can indicate what yours will be

We drove down from my parents house, did not get there at rope drop, and did not follow any plan. I didn't think it would matter much because as I said, I wasn't expecting the hoards of people that were there. But, if you get there early, follow a plan, etc you'll probably have fewer crowds
You'll have a great time whenever you go!!