In all of the written resources I have looked at regarding Jordan normal forms of matrices, the Jordan normal form $J$ is defined as having a block structure
$$ J = \begin{bmatrix} J_1 & & \\ & \ddots & \\ & & J_n \end{bmatrix} $$
where each of the $J_i$ is called a Jordan block has the structure
$$ J_i = \begin{bmatrix} \lambda_i & 1 & & \\ & \lambda_i & \ddots & \\ & & \ddots & 1 \\ & & & \lambda_i \end{bmatrix} $$
where there must be 1s on the superdiagonal. But after watching this video, it appears that what the video creator calls Jordan blocks do not need to have 1s in each element of the superdiagonal, but what the video creator calls Jordan boxes do have to have 1s in each element of the superdiagonal. I have not seen the terminology "Jordan boxes" used anywhere else, so it seems non-standard. Here is a secreenshot from the video illustrating the video creator's usage of these terms:
In symbols, the video creator's definition of a Jordan block gives it the structure
$$ J_i = \begin{bmatrix} \lambda_i & \delta_1 & & \\ & \lambda_i & \ddots & \\ & & \ddots & \delta_{m} \\ & & & \lambda_i \end{bmatrix} $$
where on the superdiagonal, $\delta_k \in \{0,1\}$. So my question boils down to this: Do all Jordan blocks need to have 1s in each element of the superdiagonal, or can there be zeros on the superdiagonal?
It's a matter of terminology. Thing is, a Jordan matrix might have different blocks with the same eigenvalue on the diagonal. So you can describe a Jordan matrix like you did at the beginning of your post, but then note that the elements $\lambda_i$ are not necessary distinct.
Now, it makes sense to give a name to the collection of all blocks which correspond to the same eigenvalue. If it was me, I would say that a Jordan block has the form:
$J_i = \begin{bmatrix} \lambda_i & 1 & & \\ & \lambda_i & \ddots & \\ & & \ddots & 1 \\ & & & \lambda_i \end{bmatrix}$
And the bigger block which we gain from combining all the blocks corresponding to $\lambda_i$ (note that between each two such blocks there is a $0$ on the superdiagonal, and so the bigger block indeed might have $0$'s there) might be called a Jordan box corresponding to $\lambda_i$. But the creator of the video decided to switch the names, he called this a Jordan block, and he called the smaller blocks (where there are only $1$'s on the superdiagonal) a Jordan box. It's just terminology. What matters is to understand how a Jordan matrix looks like.
Such things happen a lot in mathematics. Actually, things sometimes get even more confusing. Some define a Jordan block with the $1$'s being below the main diagonal, not above it. (For example, in Hoffman's book)