Correct for Part 91. But, for Part 121 it may not be the case. It certainly wasn't for my company.
I'd hope you aren't flying for hire the day after you get your IR
You are correct. For part 121, the minima are increased for restricted (less than 100hr) Captains. However, there is no hard limit on hand flying down to minima. The company wants us to use the AP if the visibility is low, but we can be dispatched without it all the way down to 200ft and 1800RVR...