Let $\mathfrak M = (M, 0, +, - , r)_{r\in R}$ be a structure which realizes a module, i.e., $(M, 0, +, -)$ is an abelian group together with operations $r : M \to M$ with $R$ a ring with $1$. Let $L_{Mod}(R) = L_{AbG} \cup \{r : r \in R \}$ with $L_{AbG} = \{0,+,-\}$ be the language of $R$-modules.
Definition: An equation is an $L_{Mod}(R)$-formula $\gamma(\overline x)$ of the form $$ r_1 x_1 + r_2 x_2 + \ldots + r_m x_m = 0. $$ A positive primitive formula ($pp$-formula) is of the form $$ \exists \overline y (\gamma_1 \land \ldots \land \gamma_n) $$ where $\gamma_i(\overline x, \overline y)$ are equations. These definitions are taken form K.Tent/M.Ziegler, A Course in Model Theory. In these notes it is mentioned that we can combine $pp$-formulas with intersection. If $\varphi(\overline x)$ and $\psi(\overline x)$ are pp-formuals then $$ (\varphi\cap \psi)(\overline x) = \varphi(\overline x) \land \psi(\overline x). $$ But why would this give a $pp$-formula? For example if $$ \varphi(x) = \exists y ( r_1 x + r_2 y = 0 ), \quad \psi(x) = \exists y ( s_1 x + s_2 y = 0 ) $$ then $$ \exists y ( r_1 x + r_2 y = 0 ) \land \exists y ( s_1 x + s_2 y = 0 ) $$ is not a pp-formula as the existential quantifier does not appears at the beginning? Maybe we could write $$ \exists y \exists z ( r_1 x + r_2 y = 0 \land s_1 x + s_2 z = 0 ) $$ but this is not what is suggested in the text...
The sentence you wrote at the end is correct. Although formally, you should maybe write $$\exists y\, \exists z\, (r_1x+r_2y+0z = 0 \land s_1x+0y+s_2z = 0)$$ because the definition says a pp-formula should have the form $\exists z\, (\gamma_1(x,z)\land \dots\land \gamma_n(x,z))$, where the $\gamma_i$ are equations in the variables $xz$, i.e. they include all of the free and quantified variables.
You say that this isn't suggested in the text, but I think it is - sometimes you have to read between the lines a little!
First, you're expected to know the basic logic fact that the conjunction of two existential formulas is equivalent to an existential formula, but you have to rename any conflicting variables: $$(\exists y\, \varphi(x,y))\land (\exists y\, \psi(x,y)) \Leftrightarrow \exists y\, \exists z\, (\varphi(x,y)\land \psi(x,z)).$$
Now the authors of the linked document write: "We can see any equation $\gamma(x,y)$ in some variables as an equation $\tilde{\gamma}(x,y,z)$ relating more variables, simply attaching zero coefficients to the new variables. Hence we can combine two pp-formulas without mixing up existentials."
So they're suggesting the following procedure: