I have seen it so far only as 'incidence density'.

However I found the following:
Incidence density - synonym for incidence rate
Incidence rate - Number of incident events divided by the cumulative at-risk time in the sample: a time-constant incidence ‘force’ (hazard)
https://link.springer.com/article/10.1007/s00134-014-3279-7

Examples:
Incidence density is defined as the number of incident cases occurring in a susceptible population followed over a given time period; its units are therefore expressed as the number of cases per unit of person time. For example, if four cases of a disease occur in a population of 1000 people followed for one year, the incidence density for this disease in this particular population would be 4 cases per 1000 person years. The incidence density ratio is defined as the ratio of incidence density of an exposed group to that of an unexposed group. In the example above, if the group described had been exposed to a particular agent and if the incidence density of a comparable unexposed group were 2 cases per 1000 person years, the incidence density ratio of these two groups would be equal to 2 (4 per 1000 person years/2 per 1000 person years). As noted above, the odds ratio has been shown to be an unbiased estimator of this incidence density ratio. (1) Consequently, there is usually no need to invoke the rare disease assumption or to consider the odds ratio as being an approximation to the rate ratio.
https://www.bmj.com/content/320/7233/503/related

The national cost of adverse events in adult inpatients was estimated as the product of (1) the estimated number of adverse events—using the INAES incidence density of adverse events applied to the number of adult inpatient admissions to acute public hospitals in 2009, excluding those with obstetric and psychiatric principal diagnoses (n=339,84434); and (2) the average cost of an event—calculated as the INAES mean number of added bed days attributed to adverse events multiplied by the average cost of an inpatient hospital bed in Ireland in 2009 (€909 per day36).
https://qualitysafety.bmj.com/content/26/2/111