runnerHaving said that while most distance runners use asics.......
Can't agree. A very high proportion of serious club distance runners use Nike, Asics comes a distant second.
Asics are also a popular shoe, particularly as they tend to be a little cheaper with sales from time to time. Nike Pegasus has an air cushion. The problem with Asics is that it tends to break down/harden with time.
WRT query on shoes for weightlifting; you need specific shoes for your activity - runners will NOT provide suitable stability for weightlifting.
Pedigree: mileage: 50-70 p.w., 15 sub 3:00s, Pb 2:46, running 23 years - first 6 with Adidas, last 17 Nike, mainly Pegasus. Occasional foray into "new super shoes" but, time and again, back to Pegasus. Largely injury free during this time - this is the acid test for a good shoe. Have used Asics for racing, mainly for lightness but find Asics racers wear quickly.