Maxi Cosi Nomad is fine. That conforms to the ECE R129 safety regulation and is allowed for children over 15 months up to a maximum of 105cm/18kg. Rear facing would be better, but this seat is legal and better than a seatbelt alone.
The second link is a type of seat which are not safety tested. They rip apart horrifically in accidents and don't provide any crash safety. They are also illegal to use. It doesn't matter which exact one it is, it's called a "No-frame" or "Frameless" car seat. They are common in countries with very lax car seat laws e.g. Asian countries, Russia. We are seeing them in the UK because of the direct import sellers who now flood sites like amazon, and advertise (for direct sites as you linked) on social media. People see it listed and assume that because amazon is a reputable company, or the website has a .co.uk ending, the product must be safe. The problem is that amazon doesn't vet every product which comes through from third party sellers.
Your child would be safer in a seatbelt than this type of seat, but neither seatbelt nor "frameless" car seat is legal. Please watch the video linked, do not use the seat (I realise it was the Maxi Cosi you used before, which is fine) and tell your friend. They shouldn't be sold at all but amazon apparently don't give a shit about this, and nobody can stop people setting up websites to sell whatever they like.
If she needs a portable seat and can't afford the Maxi Cosi Nomad, the cheapest of this type of seat but with a proper safety standard I can find currently is Kinderkraft Comfort Up which is £50 in a few places. Or Joie Tilt, £60 at Argos, or Halfords have a few at £60 as well including one rear facing seat and one forward facing with isofix. Boots have Cozy n Safe Fitzroy for just under £60 with their current offer.
I know that is not as cheap as a £20 seat - but I'd use a second hand seat if necessary over those things.