Does adding a disconnect box below the meter mean that my service panel is now a subpanel?
If the power company did not change the bonding point, you still have a main panel ("the one with the bond in it.")

So long as there's only one, the location of the bond can and does vary with the preferences of the power company, mostly. Mine is inside the meter box, so my panel is wired as a sub-panel.

On a typical panel run from a meter and bonded in the panel, the meter itself is a disconnect point that's upstream of the bond, and that's not a problem.

You should realize that the neutral and ground conductors are not interrupted by a disconnect, so the bond in your panel is still effective even with a disconnect upstream of it.

If the power company had opted to move the bond point, they would have needed to rewire your panel - a considerable expense (for them) and inconvenience (for you) they avoid by not changing the bond point.