G13 and Gmaj13 are two different chords.  And the chord finder only has G13.  But you have the right idea.

G13 is a dominant chord.  The dominant is implied in standard naming:  any chord with just a number (7, 11, 9, 13) is a dominant.  And dominant chords have a minor 7th.

1 - M3 - 5 - m7 - 9 (2) - 11 (4) - 13 (M6)

Gmaj13 has a major 7th.  That's what the "maj" indicates.  Hope that helps!