Sheree, you are correct in that Standard offered a slim and a stout model. From what I have seen, these are rotary models, the bobbin, in the modern style, rotates in the bobbin case (holder). The bobbin is shaped like a donut, short and with the hole in the center. The single plate covering the bobbin slides (or is hinged) to give access to remove and install the bobbin.
Another major type of treadle is a vibrating shuttle. The bobbin is placed in a shuttle, which is placed in a shuttle carrier. This bobbin is shaped like a long, skinny, mini spool of thread. Hence the name, long bobbin. The giveaway on this type of mechanism is that there are two slide plates, long and narrow, one slides off to the back, the other to the front.
I'm no expert on Standards, and it could be that they manufactured both kinds.