And lastly, the save on the front page still does cause lag, just to a lesser degree because of less particles.
Also, I believe you said on that front page save that lag is a delay in connection with a server. That is technically wrong. That would be latency, which yes, can CAUSE lag. However, in the case of powder toy and lag, internet connection is not a factor. Lag is impaired computer functionality (slow application responses or lower framerates) resulting from latency (not in this case), or low-preformance hardware. In non-computer situations, lag still basically means to be slow.
If you want to get technical, then yeah sure, it's 'hardware slowdown' caused by the cpu and gpu having to do calulations about the temp., speed, position, etc about the particles and then displaying them. And since no cpu/gpu is infinitly fast, it takes time. But your average person calls this phenomenon "lag" because they really don't care.