I think you're confusing the horizon of places on the surface of Earth that you can see with the distance to another satellite you can see.
6900km from the center of Earth. Figure you don't want the link to point within 150km (6500km) of Earth, to not pass through much atmosphere and to not see too much atmospheric glow (even with narrow filters, this matters).
Effectively you have an isosceles triangle with 6900km on the common side and an altitude of 6500km (tangent to "top of atmosphere" at 150km.
6900km from the center of Earth. Figure you don't want the link to point within 150km (6500km) of Earth, to not pass through much atmosphere and to not see too much atmospheric glow (even with narrow filters, this matters).
Effectively you have an isosceles triangle with 6900km on the common side and an altitude of 6500km (tangent to "top of atmosphere" at 150km.
sqrt((6900^2 - 6500^2)) * 2 =~ 4600km