Earlier quoted context omitted.
This argument is not logically valid. Nothing changes with respect to the possibility of defining something if there is no free will - defining something is merely mapping an identifier to a representation of meaning. A purely mechanical device can define things just fine. Free will is demonstrably not necessary to define things.
It is logically valid, because it’s like asking a candle flame to flicker in a certain pattern to prove it can control how it flickers to other candle flames. If we have no free will we are just like candle flames; any argument that says we are more than candle flames is not valid when you assume we are candle flames. People forget that logic requires axioms.
If I were to ask you to act a certain way and you do, this would neither prove nor disprove free will, because it tells me nothing about whether causality required you to act that way or differently and whether or not you deviated from causality, and so your analogy is meaningless. It is not what we'd ask you to do in order to prove free will.
You're using an assumption about an invalid argument as an argument in favour of free will, and that's no more valid than the original invalid argument.
To prove free will you need to prove that there are decisions in the brain that does not follow a) strict causality, b) randomness, c) any combination of the two. As you've pointed out, experiments where you ask a subject to act in specific ways do not produce evidence for it, as they can just as well just be causality in action. That the strawman hypothesis you presented is not valid does not prove the inverse, nor does it release any obligation of proponents of the idea of free will from providing evidence, or even just a definition to support their idea.
Free will is an absolutely extraordinary claim in that respect, because it implies a third category of events that is not a mix of strict cause-effect chains and random events, but introduces a "will-effect" chain that has no direct causal link but at the same time is not random.
As it stands, without a definition, the notion of free will is not even a fanciful hypothesis - to suggest it is a hypothesis would imply a fully formed idea. But the concept is so woolly and meaningless it tells us nothing.
It's at most a fervent wish or hope of not being an automaton. But a wish is not evidence, nor an argument, or even a hypothesis.
If you want to produce a valid argument for free will, you need to be able to either proffer a hypothesis for how such a "will-effect" chain can exist, or demonstrate physical transitions in a mind that are demonstrably not cause-effect or random byproducts of ordinary physics.
Even just come up with a coherent definition and you will be world famous, as you'd have introduced a new area of logic. If you can prove a physical mechanism for it, you'll have Nobel Prize in physics in the bag.