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Is there a natural intermediate version of PA?

Below, “logic” means “regular logic containing first-order logic” in the sense of Ebbinghaus/Flum/Thomas. Setup For a logic $\mathcal{L}$, let $\mathcal{PA}(\mathcal{L})$ be the set of $\mathcal{L}$-sentences in the language of arithmetic consisting of: the ordered semiring axioms, and for each $\mathcal{L}$-formula $\varphi(x,y_1,…,y_n)$ the induction instance $$\forall y_1,…,y_n[[\varphi(0,y_1,…,y_n)\wedge\forall x(\varphi(x,y_1,…,y_n)\rightarrow\varphi(x+1,y_1,…,y_n))]\rightarrow\forall x\varphi(x,y_1,…,y_n)].$$ (Note that even if the new logic […]