We discuss the origin of two triggers of Swift ’ s Burst Alert Telescope ( BAT ) that occurred in 2011 . The triggers were identified with Swift J185003.2–005627 , a previously unknown X-ray source , and the known but unclassified X-ray transient Swift J1922.7–1716 . We investigate the BAT data and follow-up observations obtained with Swift ’ s X-ray and ultraviolet/optical telescopes to demonstrate that both triggers are consistent with thermonuclear X-ray bursts . This implies that both sources are neutron star low-mass X-ray binaries . The total duration of \simeq 7 minutes and estimated energy output of \simeq ( 3 - 7 ) \times 10 ^ { 39 } erg , fall in between that of normal and intermediately long X-ray bursts . From the observed peaks of the X-ray bursts , we estimate a distance of \lesssim 3.7 kpc for Swift J185003.2–005627 and \lesssim 4.8 kpc for Swift J1922.7–1716 . We characterize the outburst and quiescent X-ray properties of the two sources . They have comparable average outburst luminosities of \simeq 10 ^ { 35 - 36 } ~ { } \mathrm { erg~ { } s } ^ { -1 } , and a quiescent luminosity equal to or lower than \simeq 2 \times 10 ^ { 32 } ~ { } \mathrm { erg~ { } s } ^ { -1 } ( 0.5–10 keV ) . Swift J185003.2–005627 returned to quiescence \simeq 20 d after its BAT trigger , while Swift J1922.7–1716 appears to exhibit long accretion outbursts that last several months to years . We identify a unique counterpart for Swift J1922.7–1716 in the ultraviolet/optical data . Finally , we serendipitously detect a flare lasting \simeq 500 s from an uncataloged X-ray/optical object that we tentatively classify as a flaring M-dwarf .