This confusion was created the other side's choice of name, whether purposely or not.

You start off with a phrase like Black Lives Matter. This is clearly a political slogan designed to provoke opponents into taking the negation of it, i.e. into claiming that black lives don't matter.

The problem with it is that it's too clever by half, because it gives opponents the ability to claim the middle. All Lives Matter.

And it's a two party system so now both the center and the right are standing under the banner of the neutral slogan. Now BLM declares that ALM are really just a bunch of secret racists because otherwise they have to eat the rhetorical self-own.

But this is just an attempt to reassert the politically-motivated original framing, i.e. that anyone who disagrees with any BLM political position -- even if the disagreement comes from the center -- is saying they think black lives don't matter. Which is not necessarily the case.