So these are all very valid and common questions, so I am hoping to shed some light onto the whole situation and let you in on a little secret (which will save you money and hassle in the end).
Old Fashioned or Rolled Oats (My personal favorite)
Old fashioned oats are the biggest of the oats. The key here is that the oats are rolled and not cut. They hold their shape and remain visible in your recipes and often result in a chewier product. My suggestion is to ONLY buy old fashioned oats when you are at the store. The reasoning for this is because you can quickly and painlessly turn old fashioned oats into quick oats with a few pulses in the food processor. As such, a batch of old fashioned oats will serve all of your baking and cooking needs.
These oats are similar to old fashioned oats but have been cut down into smaller pieces making them easier and quicker to cook. When using quick oats in a recipe, it provides for more substance but without a chewiness that comes from the larger oats.
Steel Cut Oats
The key difference to steel cut oats is that they are actually groats. Groats are oats after they have been cleaned, toasted and hulled. The groats are then cut into pieces, instead of rolled. As a result, this type of oat is much chewier and takes a longer time to cook a break down. Steel cut oats are not ideal for baking as they do not break down during the cooking process as easily as the rolled oats.
I hope this helps you with your questions and you know exactly what to grab next time you are looking to make a batch of oatmeal cookies.