In the first example there is an acting force perpendicular to the direction of the center of mass, resulting in a linear velocity acceleration in the direction of the force.
In the second example there is an acting force which is not perpendicular to the center of mass, resulting in an acceleration of linear and angular velocity.
I know what would be happening (or correct me if I don't), but not how I could calculate it?!
Tracking the center of mass $(x_C,y_C)$ and rotation $\theta$ based on the net force components $(F_x,F_y)$ and net torque about the center of mass $\tau_C$
$$ \begin{aligned} m \ddot{x}_C & = F_x \\ m \ddot{y}_C & = F_y \\ I_C \ddot{\theta} & = \tau_C \end{aligned} $$
where $I_C$ is the mass moment of inertia about the center of mass.
If the loading on the body is a single vertical force $F$ that is offset by $d$ from the center of mass, then
$$ \left. \begin{aligned} m \ddot{x}_C & = 0 \\ m \ddot{y}_C & = F \\ I_C \ddot{\theta} & = d\,F \end{aligned} \right\} \begin{aligned} \ddot{x}_C & = 0 \\ \ddot{y}_C & = \tfrac{F}{m} \\ \ddot{\theta} & = \tfrac{ d\,F}{I_C} \end{aligned} $$
You have to balance forces and moments separately.
The linear motion is governed by:
$$\Sigma F_y = ma_y$$
And the rotation is governed by:
$$ \Sigma M_O = I\alpha$$
Where you have to choose a point $O$ about which to sum the moments $M$. The moment due to a force is the force magnitude times the distance $L$ from the point $O$. So for example your moment summation would look like:
$$ LF_y = I\alpha$$
(In your case there is only one source of moment)
