EDIT: Another view of many issues of decidability is this one, borrowed and simplified from one used in complexity studies in computer science. If you have a decision or labellingproblem, where you have a set S of instances and for each instance you want to say"yes, instance I has property P" or "no, I does not have P", you take a somewhatPlatonist viewpoint and say " I will group those instance which have P into this subsetR", and then you end up with two sets, S and a proper subset R. Then you shift to aconstructivist mode and ask "Is there a way I can tell quickly, or even mechanically, whena member of S is also a member of R or not?" Then you switch to programmer/computerscientist mode and say "Let's see if I can either a) write a program to determine ifan instance is a member of R, or b) translate the domain to one where I can encodethe halting problem, so that determining membership in R solves the halting problem" .If the set R is recursive inside S, then a) is possible in theory, but may be difficult or impossible in practice, depending on the complexity of the set R. If the set R is not recursive in S, then b) may or may not be possible, but is usually the first stepone tries.

How does one show R recursive in S or not? One takes an encoding, which is aninjective and computable map from S into the natural numbers (or computably functionalequivalent), and then sees if the image of R under this map is a recursive subset ofthe natural numbers. So this and the previous paragraph are a long winded way ofsaying that most issues of decidability involve coding the problem up in a way asto move the question into the realm of subsets of natural numbers, and using recursiontheory or diagonalization or something to determine the status of the image set. Forme, I picture the set of identities or the set of terms as a set of numbers, each number colored with label or term or identity it represents, and I picture the subset withproperty P as a subset of integers which may or may not be a recursive subset. Theset of identities satisfied by the real numbers with exponentiation , addition andmultiplication is a set which has a logically equivalent, recursive, and non finitesubset. The set of terms in the Richardson theorem which are equivalent to 0 is anonrecursive subset of the set of all terms used in the context of the theorem. END EDIT

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I am not a professional logician, but I have studied mathematical logic, and in past work I used the rough notion (as have many before me) that if you can write a Pascal program to decide correctly the yes or no answer to a problem given the finite set of parameters as input, then the problem or issue is decidable. Otherwise it isn't. Taken at this level, I see both uses of decidability as the same. In one, there is a finite specification which can be used to test whether an identity is in the one set, in the other there is no such program to test whether an equation/identity is in the other set.

(There are technical arguments to be made as to which machine model, complexity, degree of undecidability if one looks at e.g. Turing equivalent degrees, and so on. I am setting aside all these complexities and ways to distinguish the two uses of decidability, since they seem to me irrelevant to the basic intent of your question.)

I can see both problems as problems of clone theory. Again roughly, the first problem talks about whether there are a finite number of relations in the generators in addition to the general relations for a clone that can be used to describe the collection of equivalence classes of terms (there are not, but there is a recursive set of such relations). The second talks about whether the set of terms in the clone equivalent to the term 0 is describable by a computer program; according to Richardson, it is not. There are other ways to recast the problems to see some similarities and highlight the differences; it depends on just what you want to see.