I undertand now, I misunderstood the definition.

And the question came to my mind when I read the Definition 11.1 in Chapter 2. It is about identities and things like $A\models p\approx q$, and there is an obvious ambiguity when the tuple $(x_1,...,x_n)$ is not clear. Also, when I went back to the proof of Theorem 10.3(c) in Chapter 2, I find it hard to show $E^{k}(S)$ is equal to RHS, because of similar problems.

I don't know is $X$ allowed to be $\{x\}$, in this case, all terms are of the the form $p(x)$, and obviously the term functions can no longer 'generate' $Sg(X)$, when considering an algebra $A$ and its subset $X$.

Yes! I mean deduction rules are (K), (MP) and (Gen)

Thanks a lot! I was reading a paper, where the logic is a little bit different, but I believe that if there is only one diamond, then that logic is the same as K. I guess the claim I made in my question is something used in that paper. Something might be wrong, the paper or my paraphrase. I hope I can find it out.