  The $\imath$Hall algebra of the projective line is by definition the twisted
semi-derived Ringel-Hall algebra of the category of $1$-periodic complexes of
coherent sheaves on the projective line. This $\imath$Hall algebra is shown to
realize the universal $q$-Onsager algebra (i.e., $\imath$quantum group of split
affine $A_1$ type) in its Drinfeld type presentation. The $\imath$Hall algebra
of the Kronecker quiver was known earlier to realize the same algebra in its
Serre type presentation. We then establish a derived equivalence which induces
an isomorphism of these two $\imath$Hall algebras, explaining the isomorphism
of the $q$-Onsager algebra under the two presentations.
