In this answer let us for simplicity only consider circular orbits. OP's eqs. (1) & (2) are closely related to Kepler's 2nd and 3rd law, respectively, and their coexistence.

The main point is that when we state a power law (or a law of proportionality), it is important to understand the limitations of the law, i.e. the assumptions that make the proportionality constant actually constant. (To see another example of what can go wrong, see e.g. this Phys.SE post.)

In OP's case: If we compare 2 different circular orbits, there is no reason why they should have the same angular momentum, i.e. in that comparison, we should discard OP's eq. (1).