Sketching lattice points, primitive axes, primitive cell, and basis of atoms

Question

I am currently studying Introduction to Solid State Physics, 8th edition, by Kittel. Chapter 1 provides the following figure and accompanying explanations:

It also says the following:

An ideal crystal is constructed by the infinite repetition of identical groups of atoms. A group is called the basis. The set of mathematical points to which the basis is attached is called the lattice. The lattice in three dimensions may be defined by three translation vectors $\mathbf{a}_1$, $\mathbf{a}_2$, $\mathbf{a}_3$, such that the arrangement of atoms in the crystal looks the same when viewed from the point $\mathbf{r}$ as when viewed from every point $\mathbf{r}^\prime$ translated by an integral multiple of the $\mathbf{a}$'s: $$\mathbf{r}^\prime = \mathbf{r} + u_1 \mathbf{a}_1 + u_2 \mathbf{a}_2 + u_3 \mathbf{a}_3. \tag{1}$$ Here $u_1, u_2, u_3$ are arbitrary integers The set of points $\mathbf{r}^\prime$ defined by (1) for all $u_1, u_2, u_3$ defines the lattice. The lattice is said to be primitive if any two points from which the atomic arrangement looks the same always satisfy (1) with a suitable choice of the integers $u_1$.

Here are some lattice points I drew:

It doesn't seem to me that any of these lattice points are primitive; that is, I don't think that any of these lattice points satisfy $\mathbf{r}^\prime = \mathbf{r} + u_1 \mathbf{a}_1 + u_2 \mathbf{a}_2 + u_3\mathbf{a}_3$. The irregular pattern present in this space lattice makes it difficult to see how this could be done.

So, in order to satisfy figure 3c, what are the lattice points of interest?

I would greatly appreciate it if people would please take the time to explain this.

EDIT

Answer 1

I think the cell after EDIT is correct. Here I added a few neighboring cells:

Admittedly, the atom in the right bottom corner is missing. Apparently the author didn't consider it important.

Answer 2

You have to take care about the definition of lattice and crystal structure. Quoting your text "An ideal crystal is constructed by the infinite repetition of identical groups of atoms. A group is called the basis. The set of mathematical points to which the basis is attached is called the lattice". So the crystal structure is constituted of a basis (an atom or a group of atoms) attached to a lattice.

By definition a lattice is an infinite array of mathematical points in space, in which each point has identical surroundings to all others. your Fig. 3a represents a lattice. Conversely your fig 3c does not represent a lattice; in fact in the figure caption it is stated that the points represent atoms. In this case case each pair of points represent your basis, not lattice points. You're asked to find the lattice points. The cell after EDIT is correct; in that case lattice points are at the same position of one of the two points forming the pair.