Well the Lenovo ain't a gaming laptop as its using HD Graphics 505 which classes it as a office laptop, also can't find the Aspire e5 one anywhere (post a link to the pages)
Also there are no 'cheap' gaming laptops. If you want a gaming laptop your going to have to pay £1000+. Anything that is compact and powerful cost more money. Your going to at least want a GTX 1060 in the laptop but the moment you add that in the price shoots up to £1,300/400.
Also with a laptop in this current gen of tech and games an SSD is a must really unless you like waiting around a lot. This shoots the price up even higher.
I have used Integrated graphics back in 2014 on a dead cheap Medion laptop that had an Haswell i3 & Intel HD graphics 3000.... it...