Unfortunately, it is more complicated than that. Things change when it gets to the retail store. At the mill, the boards are generally cut with a very large and aggressive blade, so they come off the saw with a rough-sawn surface. Many hardwood sellers decide to surface the wood smooth. Generally, they can charge more for a smooth board, but it also helps the buyer. It is very difficult to see grain patterns and color in the wood when it is left rough-sawn. Additionally, retailers are limited to selling their boards to woodworkers who have planers and are willing to do the extra work of surfacing the lumber. But if you plane your own rough-sawn lumber, you may be able to get thicker boards.
Most, hardwood dealers offer the option of surfacing the lumber before selling to the consumer. They sell their lumber as either S1S, which means surfaced one side or S2S, surfaced two sides.
Boards lose some thickness during the surfacing process (a slight amount is lost in the drying process, as well, so a four quarter, S1S lumber is about 7/8" thick, and S2S lumber measures just 13/16" thick. However, it is still referred to as four-quarter—probably for simplicity.
When I go to the hardwood store I ask for four-quarter stock when I need a board for a project calling for ¾" lumber. I ask for five-quarter when I need a 1" piece. I generally ask for S2S because I have a small planer, and I want to prevent some wear and tear, but generally there are options. My experience is hardwood sales reps are very friendly and willing to assist if you need it. (Reference SHOPNOTES Magazine Vol 36/No.216)