In Goldstein's Classical Mechanics (2nd ed.), Section 9-1, pgs. 382-385, the generating functions (hereafter denoted $F$) for canonical transformations are introduced. From here on out, I'll refer to the original canonical coordinate variable and canonical momenta variable as $q$ and $p$, while the "new" canonical coordinate and momenta variables which I'm transforming to will be $Q$ and $P$. I believe this is standard notation.

There are four separate cases enumerated in the text - when the canonical transformation is best described by a generating function dependent on only on the mixed coordinates:

1. $q$ and $Q$ ~ the transformation relations can be found by setting $F=F_1(q,Q,t)$
2. $q$ and $P$ ~ the transformation relations can be found by setting $F=F_2(q,P,t)-QP$
3. $p$ and $Q$ ~ the transformation relations can be found by setting $F=F_3(p,Q,t)+qp$
4. $p$ and $P$ ~ the transformation relations can be found by setting $F=F_4(p,P,t)-QP+qp$

From what I've read online, the four types of variable-dependencies for the generating function enumerated above are known as the four-kinds of the generating function (i.e. it seems well-defined as standard nomenclature in the literature).

My question is, how come the cases where the canonical transformations are best described by a generating function of non-mixed coordinates covered? (i.e. $F_5(q,p,t)$ or $F_6(Q,P,t)$) Are there no canonical transformations best described by such generating functions? Am I overseeing something trivial?

For example, suppose there was a canonical transformation that was best described by a generating function dependent only on $q$, $p$, and $t$. Then for the generating function I could use

$$F=F_5(q,p,t)-pQ$$

I could plug that into the fundamental relation between the original and newly transformed-to quantities to then get the transformation relations.

$$\begin{align*}\require{\cancel} p\dot{q}-H&=P\dot{Q}-K+\frac{dF}{dt}\\ &=P\dot{Q}-K+\frac{\partial F_5}{\partial t}+\frac{\partial F_5}{\partial q}\dot{q} +\frac{\partial F_5}{\partial p}\dot{p} - \dot{p}Q - p\dot{Q}\\ &\\ &\rightarrow \left(H-K+\frac{\partial F_5}{\partial t}\right)+\left(\frac{\partial F_5}{\partial q}-p\right)\dot{q}+\left(P-p\right)\dot{Q}+\left(\frac{\partial F_5}{\partial p}-Q\right)\dot{p}=0 \end{align*}$$

I can now read off the transformation relations as:

$$Q=\frac{\partial F_5}{\partial p}\tag{1}$$

$$p=\frac{\partial F_5}{\partial q}\tag{2}$$

$$P=p\tag{3}$$

$$K=H+\frac{\partial F_5}{\partial t}\tag{4}$$

I can see that (1) and (3) define the canonical transformation, whereas (2) puts a restriction on what functions $F_5(q,p,t)$ will work. Nevertheless, this still defines a family of canonical transformations, doesn't it?

Background.

In Goldstein's Classical Mechanics (2nd ed.), Section 9-1, pgs. 382-385, the generating functions (hereafter denoted $F$) for canonical transformations are introduced. From here on out, I'll refer to the original canonical coordinate variable and canonical momenta variable as $q$ and $p$, while the "new" canonical coordinate and momenta variables which I'm transforming to will be $Q$ and $P$. I believe this is standard notation.

There are four separate cases enumerated in the text - when the canonical transformation is best described by a generating function dependent on only on the mixed coordinates:

1. $q$ and $Q$ ~ the transformation relations can be found by setting $F=F_1(q,Q,t)$
2. $q$ and $P$ ~ the transformation relations can be found by setting $F=F_2(q,P,t)-QP$
3. $p$ and $Q$ ~ the transformation relations can be found by setting $F=F_3(p,Q,t)+qp$
4. $p$ and $P$ ~ the transformation relations can be found by setting $F=F_4(p,P,t)-QP+qp$

From what I've read online, the four types of variable-dependencies for the generating function enumerated above are known as the four-kinds of the generating function (i.e. it seems well-defined as standard nomenclature in the literature).

My question is, how come the cases where the canonical transformations are best described by a generating function of non-mixed coordinates covered? (i.e. $F_5(q,p,t)$ or $F_6(Q,P,t)$) Are there no canonical transformations best described by such generating functions? Am I overseeing something trivial?

Example.

Suppose there was a canonical transformation that was best described by a generating function dependent only on $q$, $p$, and $t$. Then for the generating function I could use

$$F=F_5(q,p,t)-pQ$$

I could plug that into the fundamental relation between the original and newly transformed-to quantities to then get the transformation relations.

$$\begin{align*}\require{\cancel} p\dot{q}-H&=P\dot{Q}-K+\frac{dF}{dt}\\ &=P\dot{Q}-K+\frac{\partial F_5}{\partial t}+\frac{\partial F_5}{\partial q}\dot{q} +\frac{\partial F_5}{\partial p}\dot{p} - \dot{p}Q - p\dot{Q}\\ &\\ &\rightarrow \left(H-K+\frac{\partial F_5}{\partial t}\right)+\left(\frac{\partial F_5}{\partial q}-p\right)\dot{q}+\left(P-p\right)\dot{Q}+\left(\frac{\partial F_5}{\partial p}-Q\right)\dot{p}=0 \end{align*}$$

I can now read off the transformation relations as:

$$Q=\frac{\partial F_5}{\partial p}\tag{1}$$

$$p=\frac{\partial F_5}{\partial q}\tag{2}$$

$$P=p\tag{3}$$

$$K=H+\frac{\partial F_5}{\partial t}\tag{4}$$

I can see that (1) and (3) define the canonical transformation, whereas (2) puts a restriction on what functions $F_5(q,p,t)$ will work. Nevertheless, this still defines a family of canonical transformations, doesn't it?