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. 

<h2>Problems with the given definition</h2>

1. 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 $1-3 = -2$.  

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.

3.  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 $d_i \leq 1$.

4.  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 $[0, 1]$.  

<h2>An alternate definition</h2>

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).