What is the difference between Integer Quadratic Programming versus Mixed Integer Quadratic Programming?

Question

I am new to the optimization problem of Quadratic programming. In equation $8$ of this paper, there is an equation:

$$\min \operatorname{cost} (x) = x' H x +c' H x \quad \text{subject to,} \tag{8} \\ Ax=b, \quad x\in\{0,1\}^n$$

The authors state that this is an 'Integer Quadratic Programming (IQP)' formula.

Alternatively, in this other website, there is the following equation which is described as a 'Mixed Integer Quadratic Programming (MIQP)' formulation:

$$\min\limits_x f(x) = \frac12 x^T F x +c^T x$$

From my perspective, both of the equations shown above are similar, with the only difference being that the MIQP formula has '$1/2$' included in it.

1. I am looking for an explanation on the differences between the IQP and MIQP and in particular what the '$1/2$' represents in MIQP.

2. In addition, I am interested to apply quadratic programming to the assignment problem, thus, looking for any insight into which should be used (i.e., IQP vs. MIQP) and when.