The administration and maintenance of Microsoft Exchange Server to ensure secure, reliable, and efficient email and collaboration services across an organization.
Thought it's a dead thread, but I've seen new comments popping up, so I'm going to jump in.
With built-in tools, there's no way to automatically apply different email signatures for internal and external email, so the only thing you can do is design email separate email signatures in Outlook and apply diffferent versions manually.
Another option is using third-party email signature management tools - they allow you to centrally manage email signatures for everyone and add different versions for internal and external email (they can be applied based on more specific conditions as well. There are multiple products that can do this, you can find them, for example, on the Microsoft Appsource listing: https://appsource.microsoft.com/en-us/