Apparently Logic doesn't use some kind of pre-play buffering. The workaround is to have your entire song start e.g. one bar later, at 2 1 1 1, and let Logic play from 1 1 1 1. This workaround obviously doesn't work if you've set up a cycle - in which case Logic will miss the first samples when it starts playing as well. Next time it loops through the cycle it does play properly. There's no workaround for this - it's a serious shortcoming of the way Logic handles audio playback. With cycles, you could resort to using the Pause instead of the Play button/command: this will let you start playback from somewhere before the start of the cycle instead of starting at cycle-start (which is what Play does).