If you are creating a new preset and add a drive block, it will load with the default drive setting (Rat Dist). You can then select "My drive" in the blocks library to apply the settings. You must then save the preset. From then on, that preset will load with "My drive" on your drive block. You can apply "My drive" to any preset you want that contains a drive block.
Unfortunately, you are correct regarding the name not appearing in the block library. While you are in the edit process, the name will show. But, after you save the preset and switch between presets, the names no longer appear. This is very frustrating. It would great if you could select the effect and it shows you what setting you have applied...but it doesn't. I have not found any way to verify which settings are applied. To compensate I use an excel spreadsheet to keep track of my block settings and which presets use them. To keep it straight, I use a naming convention of: Effect_DateLetter (examples: DRV_20191225a, DRV_20191225b).