I would like to ask how to cartographically represent active faults, without using geological symbols. I don't want to make a geological map, that's why I don't use that kind of symbology. I don't have the ages of faults, but I do have the type. I saw how the United States Geological Survey (USGS) made similar maps, but they used "age" instead of "type" (of faults).
Active faults are generally represented with a continuous line. You could use different colours for the active and inactive faults but the same linetype (e.g. continuous red line for active and continuous black line for inactive). Also fault extensions or possible faults are represented with a dashed line. In the case that you want to create a greyscale map and not a coloured one, using different linetypes is the only way. If you want to play by the book, you should use a linetype that symbolizes the hanging and foot walls of the fault (but since you do not need the geological information you can stick with the aforementioned).
Regarding the USGS symbology you mention, you could also group the faults by ages, since the inactive faults are characterised so based on the geological period they gave their last movement. If I remember correctly a fault is regarded as inactive if it was last active during the Neogene System.