The beneficial effects of exercise are well documented on cognitive ability (1, 2) and reducing cognitive decline (1). However, has the type of exercise ever been analysed?
In healthy adults, is there a difference on the effects of aerobic (running) and resistance training (lifting weights) on cognitive ability?
I tried to do a little research myself and like you observed, there is not much research being done on adults relating to this topic.
What I did find was research done by M.B. Pontifex et al. (2009). Although this research was done on healthy 21 year olds (Young Adults).
They researched the effects of aerobic exercise and resistance excercise on reaction time of tasks as well as accuracy of tasks amongst other things.
This data was collected via the following method:
Twenty-one young adult participants completed a cardiorespiratory fitness test and maximal strength tests. On subsequent days, task performance measures of reaction time (RT) and accuracy were collected while participants completed a modified Sternberg working memory task before the start of, immediately after, and 30 min after an intervention consisting of 30 min of either resistance or aerobic exercise and a seated rest control.
Their observation was that after aerobic excercise their response time latency (reaction time) was decreased directly after or 30 minutes after the excercise. This was not observed after or 30 minutes after doing resistance excercise.
They stated:
The current findings indicated shorter RT latency during a working memory task that was performed immediately and 30 min after an acute bout of aerobic exercise, relative to the pretest. Similar effects were not observed after acute resistance exercise or seated rest, indicating that different modes of exercise have differential effects on the executive control of working memory. Further, shorter RT latency was observed for task conditions requiring increased working memory capacity after aerobic exercise, relative to the pretest, providing support for the view that changes in cognitive function after acute exercise are disproportionately larger for tasks requiring greater amounts of executive control.
According to "Exercise, brain, and cognition across the life span" which was cited in the original question, there are different effects that differ for various populations.
Developing Brains
There is no research on the effects resistance training in children. However, general exercise has significant benefits:
a meta-analysis that aggregated results across 44 studies found an overall effect size of 0.32 for the association between childhood physical activity and fitness and cognition, with significant effects across a range of abilities, such as perceptual skills (0.49), creativity and concentration (0.40), academic readiness (0.39) and achievement (0.30), IQ (0.34), and math (0.20) and verbal (0.17) tests
Aerobic exercise has specifically been shown to increase the capabilities of relational memory and accuracy of cognitive performance.
Adult Brains
Since young brains are relatively stable, little research has been done on the effects of exercise on this population. All that has been determined behaviourly is:
aerobic fitness effects on behavior may only emerge in this high-functioning group when the task is extremely difficult or that young adults have a greater range of compensatory strategies compared with children and older adults to achieve enhanced performance
Ageing Brains
Aerobic training has many positive effects on ageing across a broad range of intensities, however resistance training appears to only have benefits with higher resistance levels.