I have two slightly different definitions for Gröbner-Bases.
1.Definition from book
Let $I$ be an ideal and $G=(g_1,\ldots,g_s)$ a basis for $I$. $G$ is called a Gröbner-Basis if $\langle LT(g_1),\ldots,LT(g_s) \rangle = \langle LT(I) \rangle$ where $LT(I) = \{cx^{\alpha}:\; \text{there exists}\; f\in I\;\colon\; LT(f)=cx^{\alpha}\}$
2.Definition in lecture
Let $I$ be an ideal and $G=(g_1,\ldots,g_s)$ a basis for $I$. $G$ is called a Gröbner-Basis if $\langle LM(g_1),\ldots,LM(g_s) \rangle = LM(I)$ where $LM(I)=\{LM(f)\;\colon\;0\neq f \in I\}$
LM=Leading Monomial, LT=Leading Term
I am not so much confused that in my lecture $LM$ is used instead of $LT$, but rather that in my lecture there is "just" $LM(I)$ and not the ideal generated by $LM(I)$. Why is that ?
The set $\mathbb{M}$ of all monomials forms a semigroup, and the set of leading monomials $LM(I)$ is a semigroup ideal in $\mathbb{M}$. The ideal basis $G=(g_1,\ldots,g_s)$ is a Gröbner basis of $I=\langle g_1,\ldots,g_s\rangle$ iff $LM(I)$ is generated as a semigroup ideal by $LM(g_1), \ldots, LM(g_s)$.
The two definitions are equivalent.