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The equitable presentation of the quantum algebra $U_q(\widehat{sl_2})$ isconsidered. This presentation was originally introduced by T. Ito and P.Terwilliger. In this paper, following Terwilliger's recent works the(nonstandard) positive part of $U_q(\widehat{sl_2})$ of equitable type$U_q^{IT,+}$ and its second realization (current algebra) $U_q^{T,+}$ areintroduced and studied. A presentation for $U_q^{T,+}$ is given in terms of aK-operator satisfying a Freidel-Maillet type equation and a condition on itsquantum determinant. Realizations of the K-operator in terms of Ding-FrenkelL-operators are considered, from which an explicit injective homomorphism from$U_q^{T,+}$ to a subalgebra of Drinfeld's second realization (current algebra)of $U_q(\widehat{sl_2})$ is derived, and the comodule algebra structure of$U_q^{T,+}$ is characterized. The central extension of $U_q^{T,+}$ and itsrelation with Drinfeld's second realization of $U_q(\widehat{gl_2})$ is alsodescribed using the framework of Freidel-Maillet algebras.

On the second realization for the positive part of $U_q(\widehat{sl_2})$ of equitable type