Why does my waveform start ahead of the trigger?

You probably have some sort of trigger filtering or delay turned on. Otherwise, a regular rising edge trigger should have caught the first pulse, not waited until the second. Look carefully thru the trigger menu and turn off anything called a "filter", "delay", "holdoff", and the like.

This same signal triggers just fine using a Tektronix TDS3014, with trigger hold off values ranging from 350μs to 1.5ms or more.

I think the answer, therefore, is to cross-check signals with more than one scope when something doesn't look right.

I was never able to get the BK 2542B to properly show the clock without either missing the first pulse, or showing random "glitching" or "garbage" making the clock pulses uncountable.