So the definition I learned is that a set $G = \{g_1,...g_t\}$ is a Groebner basis for an ideal I if $<Lt(g_1),...Lt(g_t)> = <LT(I)>$
However I'm not sure if it also requires the set G to be a normal basis for the ideal I. In most sources I read they don't mention this is required, but isisf this is the case I run into problems with reduced Groebner bases.
For example (taken from Ideals, Varieties and Algorithms, paragraph 2.7 example 1) take I = $<x^3-2xy,x^2y-2y^2+x>$ with graded lexicographic order. This has reduced Groebner basis $H = \{x^2,xy,y^2-x/2\}$
However then why would $J = \{x^2,xy,y^2\}$ not be a reduced Groebner basis too? J has the exact same leading terms as H, so should satisfy the definition just as well. However this logic makes a mistake somewhere since reduced Groebner basis are unique, but I don't see where.
In Ideals, Varieties and Algorithms by Cox, Little & O'Shea, it is required that $$ \DeclareMathOperator{\lt}{lt}\bigl\langle \lt(g_1),\dots,\lt(g_t)\bigr\rangle=\bigl\langle \lt(I)\bigr\rangle, $$ and it is proved that a Groebner basis of $I$ generates the ideal $I$ (chapter 2, §5, Corollary 6).