So the OP's question is really "what qualifies as a solo century?" Since this is really asking us how we label our own rides, this subject is therefore completely subjective, which means that arguing (asserting I'm right, you're wrong) is going to be pointless and dull. I do think there's an interesting discussion possible about how we label our own rides and why.
For me, it's pretty simple. The halfway stop is a reward to myself that helps motivate me to ride that day. Calling the ride a century no matter where and how long I take my breaks motivates me to put in a few more miles than I would if I didn't call the 100 miles a century. Unless you're in training for something specific, I seriously doubt it makes much of a fitness difference if you do the endurance ride in 4 25 mile segments in the same day vs. taking a few short breaks during a continuous 100 mile ride, all other things being equal. You can ride either one with more or less intensity than the other, do some of you define the "solo century" to motivate yourself to more intensity?