I've done a project like this and in my opinion, the way to achieve what you're wanting to do is:
1. Dye the leather the color you want the logo part to be... Let dry completely!
2. After your leather is dry, DAMPEN and press design into coaster with plastic die... let dry for about 2 hours to get nice impression... Your leather should be cool to the touch but not dry. This is explained in step 4...
3. Once you remove the die, you'll have a solid colored coaster with your nice clean logo impression in the color you want.
4. Now, with the project cool to the touch but not dry, you'll need to put something fairly heavy on it to make it layout flat. This is VERY important for the next step to work...
5. Now, take a foam roller from your local hardware store. You want the kind with NO TEXTURE! If it has texture, the wool will push down into your impressions and you'll ruin your project. Lowe's has one in their paint department for about $6 - $8 bucks.
6. Roll your roller into your complimenting dye, and roll off any extra onto a a blotter of some sort (newspaper, paper towels, etc.) then simply roll the roller over the face of your project and the dye will coat the raised parts while the depressed areas remain the other color.
7. Let dry then seal with your favorite sealer! I like Satin Sheen because you see the design before the shine...
Here's a sample of what I'm talking about.
The Red & White areas are done using this technique, the the Black area was sewn on.