The equitable presentation of the quantum algebra Uq(sl 2) is considered. This presentation was originally introduced by T. Ito and P. Terwilliger. In this paper, following Terwilliger's recent works the (nonstandard) positive part of Uq(sl 2) of equitable type U IT,+ q and its second realization (current algebra) U T,+ q are introduced and studied. A presentation for U T,+ q is given in terms of a K-operator satisfying a Freidel-Maillet type equation and a condition on its quantum determinant. Realizations of the K-operator in terms of Ding-Frenkel L-operators are considered, from which an explicit injective homomorphism from U T,+ q to a subalgebra of Drinfeld's second realization (current algebra) of Uq(sl 2) is derived, and the comodule algebra structure of U T,+ q is characterized. The central extension of U T,+ q and its relation with Drinfeld's second realization of Uq(gl 2) is also described using the framework of Freidel-Maillet algebras.