Backtick expressions work exactly like lambdas, except that they are bound to the instance they are created in every time that class is used to create one. To illustrate, ...
First, if there is a useful procedure I am strongly against using backticks because (1) it's been used in the past with an entirely different meaning and (2) it looks ugly and is not visually suggestive at all of what it does, especially not the subtle difference between other function definitions.
Second, I don't understand exactly what this difference or why it would be useful. It would help for you to give examples comparing lambda and this variation.
Third, you mention using ^ in "explicit" expressions to refer to parameters of the "created function" and I do not know what function you are referring to or what the exact semantics of this are. Again, a comparison of two expressions with and without that ^ would help. An expression is not a function and not all expressions are written inside functions. (And as to the specific proposed syntax, there already is the ^ xor operator and the most expected meaning of ^value is ~value. just as the unary + and - operators corresponds to the binary operators.
The only thing that I can think of is that you want `foo + ^bar` to be another way of writing lambda bar: foo + bar with some
under-specified behavior for evaluating foo and different under-specified behavior for evaluating bar.
Finally, if there is some other useful semantics for references inside a function definition, then I would think the best way to do that is to implement that, not add a new function difference. For example,