I managed to fix the problem, finally.
It appears that wine-preloaded had crashed, and the process was never killed. So it was still running in the background, and thus prevented running the game again.
I opened up Activity Monitor, and clicked on Process Name to sort the items by the name of the process. I looked for wine-preloaded, and saw that one of them was in red, and marked as "(Not Responding)". I clicked on that line, and then clicked the left-most button on the button toolbar (it looks like an x in a circle). It asked me what I wanted to do, and I selected "Force Quit", which made it disappear. After that, I ran Tequila, and was able to start CoH again. Happy day!