The first thing I will tell you is that I have a fair amount of fine lines and plenty of crows feet, so I understand the goal.
First, you want a good moisturizer. I use a moisturizing retinal day cream under and around my eyes before any cosmetics. This will help to plump or fill a lot of the fine lines there. Always allow it to dry or soak in completely. Wait 20 minutes to ensure the moisturizer is absorbed before adding any cosmetics.
Never swiped or wipe a color corrector under your eyes it is best to lightly dab. Using a beauty blender, or even the cheap wedge-shaped dollar store sponges work just as well! Don't press and twist either, just dab in light bouncing motion. This should provide a crisp corrected area around your eyes.