A quick way to make this precise is to says that a 1-groupoid is a Kan complex which is equivalent (homotopy equivalent) to the nerve of a groupoid: a 2-coskeletal Kan complex. More abstractly this is a 1-truncated ∞-groupoid.
More generally and more vaguely: Fix a meaning of -groupoid, however weak or strict you wish. Then a -groupoid is an -groupoid such that all parallel pairs of -morphisms are equivalent for . Thus, up to equivalence, there is no point in mentioning anything beyond -morphisms, except whether two given parallel -morphisms are equivalent. If you rephrase equivalence of -morphisms as equality, which gives the same result up to equivalence, then all that is left in this definition is a groupoid. Thus one may also say that a -groupoid is simply a groupoid.
The point of all this is simply to fill in the general concept of -groupoid; nobody thinks of -groupoids as a concept in their own right except simply as groupoids. Compare -category and -poset, which are defined on the same basis.
h-level 2 | 0-truncated | discrete space | 0-groupoid/set | sheaf | h-set h-level 3 | 1-truncated | homotopy 1-type | 1-groupoid/groupoid | (2,1)-sheaf/stack | h-groupoid h-level 4 | 2-truncated | homotopy 2-type | 2-groupoid | | h-2-groupoid h-level 5 | 3-truncated | homotopy 3-type | 3-groupoid | | h-3-groupoid h-level | -truncated | homotopy n-type | n-groupoid | | h--groupoid | h-level | untruncated | homotopy type | ∞-groupoid | (∞,1)-sheaf/∞-stack | h--groupoid