I believe in your equations $u_k (\vec{r})$ has different meanings in each equation:
* In equation 1, $u_k (\vec{r})$ refers to a periodic function in the Bravais lattice.
* In equation 2, it refers to the wavefunction $\phi(\vec{r})$ around each lattice site (it is written as $\phi(\vec{r}-\vec{R}_j)$ becasue $\phi(\vec{r})$ is centered at the origin).
Now, let's try to see if both forms are equivalent:
$$ \tag{1} \exp(i\vec{k}\cdot\vec{r})u_k (\vec{r}) \equiv^? \sum_j \exp(i\vec{k}\cdot\vec R_j)\phi (\vec{r}-\vec{R}_j)$$
Take the RHS:
$$ \tag{2} \sum_j \exp(i\vec{k}\cdot\vec R_j)\phi (\vec{r}-\vec{R}_j)= e^{i\vec{k}\cdot \vec{r}} \left (e^{-i\vec{k}\cdot \vec{r}} \sum_j e^{i\vec{k}\cdot\vec R_j}\phi (\vec{r}-\vec{R}_j) \right ) $$
Note that I only multiplied by $1=e^{i\vec{k}\cdot \vec{r}} e^{-i\vec{k}\cdot \vec{r}}$, then we must ask: are the terms in parenthesis a periodic function in the Bravais lattice?
Let's use the substitution $\vec{r}\rightarrow\vec{r}+\vec{R}_k$, where $\vec{R}_k$ is an arbitrary lattice vector. Then we have:
$$ e^{-i\vec{k}\cdot(\vec{r}+\vec{R}_k)} \sum_j e^{i\vec{k}\cdot\vec R_j}\phi (\vec{r}+\vec{R}_k-\vec{R}_j) =e^{-i\vec{k} \cdot \vec{r}}\sum_j e^{i\vec{k}\cdot(\vec{R}_j-\vec{R}_k)} \phi(\vec{r}+\vec{R}_k-\vec{R}_j) $$
Now, we can write our question as:
$$ \sum_j e^{i\vec{k}\cdot\vec R_j}\phi (\vec{r}-\vec{R}_j) \equiv^? \sum_j e^{i\vec{k}\cdot(\vec{R}_j-\vec{R}_k)} \phi(\vec{r}+\vec{R}_k-\vec{R}_j) $$
If we have an infinite lattice, then these equations are equivalent, so we know the part in parenthesis in equation (2) is periodic. 804
If instead, we have periodic boundary conditions (for example, a 2D lattice wrapped into a torus), there are some terms in the LHS that don't appear in the RHS and vice-versa. In this case, however, they would cancel out, since these terms would be separated by a distance equivalent to the boundary conditions.
This means your equation 2. can be written as:
$$e^{i\vec{k}\cdot \vec{r}} \left (e^{-i\vec{k}\cdot \vec{r}} \sum_j e^{i\vec{k}\cdot\vec R_j}\phi (\vec{r}-\vec{R}j) \right ) = e^{i\vec{k}\cdot \vec{r}} u\vec{k}(\vec{r}) $$
where $u_\vec{k}(\vec{r}) = e^{-i\vec{k}\cdot \vec{r}} \sum_j e^{i\vec{k}\cdot\vec R_j}\phi (\vec{r}-\vec{R}_j)$ is a periodic function in the Bravais lattice.
A great reference is: https://web.science.uu.nl/ITF/teaching/2014/2014vanMiert.pdf 804