I think at the very least you should require that $d_i \leq 1$ even if this is not clear from the literature.
If you don't require that $d_i \leq 1$, then lots of things don't work.
Problems with the given definition
- The definition of log canonical you gave is not independent of the choice of resolution.
Let me give an example. Set $X = \mathbb{A}^2$ and let $D = 3 \text{div}(x)$. Then $(X, D)$ is its own log resolution and so using your definition, the $b_j$'s are all zero and so this pair is log canonical. However, if $Y \to X$ is the blowup of the origin, then there is one exceptional divisor $E$ and $E$'s coefficient in $K_Y - f^*(K_X + D)$ is equal to $1-3 = -2$.
If you require that your condition holds for all log resolutions (or even all birational maps $f : Y \to X$ with $Y$ normal) then that implicitly guarantees that the coefficients of $d_i$ are all $\leq 1$. Just blow up the points on $D_i$ with coefficients $> 1$ repeatedly. Note that in Kollár-Mori, they do require all resolutions / valuations.
One really wants to say that $(X, D)$ is log canonical if and only if $(Y, -K_Y + \pi^*(K_X + D))$ is log canonical for any proper birational map $f : Y \to X$ with $Y$ normal. This also guarantees that $d_i \leq 1$.
You need both $d_i \leq 1$ and $d_i \geq 0$ in order to guarantee the Kodaira-type vanishing theorems you rely on. For instance, if $(X, D)$ is LC and if $L$ is a line bundle such that $L-K_X-D$ is ample, then $H^i(X, L) = 0$ for $i > 0$ if your $d_i$ are in $[0, 1]$.
An alternate definition
Let me say that I prefer a slightly different way to setup the definition. Write
$$K_Y = f^*(K_X + D) + \sum_j b_j E_j$$
without subtracting off the strict transform.
First notice that this means that some of the $E_j$ will be non-exceptional (that's totally ok). Then
Definition $(X, D)$ is log canonical if the $b_j \geq -1$. $(X, D)$ is KLT if the $b_j > -1$.
This definition then directly forces the $d_j \leq 1$ (respectively $d_j < 1$ for KLT). Setup this way, everything is completely independent of the choice of resolution.
Note that I didn't require that the $d_j \geq 0$ (this depends on the application, but based on 3. above, it might be reasonable to allow negative $d_i$ in the definitions. I should note that allowing negative $d_i$ is probably slightly more problematic for LC than it is for KLT, but I've written enough already).