The task, and thus difference between descriptions is essentially one of risk assessment.
If the current situation involves a higher level of risk than normal, e.g. engine fire, then some additional risk can be accepted for an immediate landing.
However, this judgement should not disregard higher risks in the attempt to land, e.g. you could accept a reduced landing distance safety margin (short runway), but perhaps not exceed the recommended crosswind. In the first instance the pilot retains control of the landing situation whereas in the second s/he probably does not. Caution, we all think that we are better than we actually are – particularly when stressed by a serious emergency.
For situations involving minimal additional risk for continued flight, e.g. single engine flight after engine shutdown / surge, then no additional risk need be considered in planning to land at a nearby airport. Also, within a small margin, not necessarily the closest airport could be chosen. The choice must be justifiable after the fact (what if) and thus might exclude an ‘economic’ choice; although such a choice coupled with fuel burn / weight reduction could be acceptable. Again beware of human bias – wish think; justifying something to yourself is often easy, but it is not the same as having to explain the decision to an accident investigator.