The definition of a lattice requires basis vectors that are linearly independent.
Why?
For example, the following three vectors are linearly independent and form the basis of a lattice:
\begin{array}{ccc} 0 & 0 & 1 \\ 0 & 2 & -2 \\ 1 & -2 & 1 \end{array}
But what if we add a fourth vector such that they're not linearly independent anymore. For example:
\begin{array}{ccc} 0 & 0 & 1 & 4\\ 0 & 2 & -2 & 2\\ 1 & -2 & 1 & 3\end{array}
Are the four vectors the basis of a lattice? Why or why not? And, if it is, why does the definition require linear independence? Is there an equivalent basis that is linearly independent?
Maybe your definition of a lattice is stated as such in order to keep terms as reduced as possible. Your original lattice obviously does not include every integer vector, but the addition of $\pmatrix{1 & 1 & 1}^\top$ does indeed "fill out" all the integer points of the lattice, as the following attempts to show.
Let us column reduce using only integer operations ($C_4 \leftarrow C_4 - C_3$ means column $4$ gets $1$ of column $3$ subtracted): $$\pmatrix{0 & 0 & 1 & 1 \\0 & 2 & -2 & 1 \\1 & -2 & 1 & 1\\ } \overset{C_4 \leftarrow C_4 - C_3}{\longrightarrow} \pmatrix{0 & 0 & 1 & 0 \\0 & 2 & -2 & 3 \\1 & -2 & 1 & 0\\ }$$
$$\pmatrix{0 & 0 & 1 & 0 \\0 & 2 & -2 & 3 \\1 & -2 & 1 & 0\\ } \overset{C_4 \leftarrow C_4 - C_2}{\longrightarrow} \pmatrix{0 & 0 & 1 & 0 \\0 & 2 & -2 & 1 \\1 & -2 & 1 & 2\\ }$$
Now it is apparent that if the third, fourth, and first columns are chosen as the basis that the new lattice has a determinant of one, thus any integer vector is within the span. You can see this from the lower triangular form here, as it has all ones along the diagonal: $$\pmatrix{1 & 0 & 0 \\ -2 & 1 & 0 \\ 1 & 2 & 1 }$$
Any integer matrix with a determinant of $\pm1$ has an integer inverse. That means all integer vectors are within its span, using only integer combinations of its columns. Thus your new lattice with the additional vector is the lattice of all integer vectors.
Since the definition of a lattice uses all integer combinations of the basis vectors, your new lattice as defined by four columns of three elements is valid, but it is not the most reduced basis to use. It would be an obfuscated form, as the identity itself is also a valid basis to use in this example. The identity would be the best basis to use here, unless of course you want to obfuscate the form for some reason.