"Has designers who consciously adopted significant ideas from" seems good enough for me.

Indeed, the definition admits that we could have two almost identical languages but only one of them is based on lambda calculus. I don't think it is a reasonable way to interpret "is based on", it requires too much knowledge of history. It is more proper to have a technical definition that is rooted in the language itself.

To put it pithily, "This popular notion is provided 'as is', without warranty of any kind, express or implied, including but not limited to fitness for a particular purpose." If you really want a property that's fully intrinsic, I'd suggest disregarding fuzzy 'based on'-ness and instead considering some particular aspect you care about, e.g., "How difficult is it to express such-and-such a technique in this language?"