I had no luck chaining APC units together (I also have a belkin but I haven't tested it). There is a sensitivity adjustment on them, but even at the lowest sensitivity I couldn't get it work. Older ones might be different, but I think currently only the APC smart UPS have true sine wave. During normal operation it just passes the regular line voltage straight through. Not until it's on backup power does it change to MSW. So if the first one is plugged into an outlet it would probably work, but as soon as you lose power both of them would stop charging.
Most of your small home/office UPS's are this offline type, as you said they just pass line power through normally. They will have a set high and low voltage threshold where they will switch to using the inverter and battery. If you want something that is always using the inverter and therefore always conditioning the incoming power you look for the "online" type.
The difference is cost, the online needs the battery charge circuit to be able to carry the whole load plus extra to recharge the battery, the offline type has a small battery charger just sufficient to charge the battery.