Firstly, no one else's insurance can write off your car for financial reasons. You are entitled to have it repaired (even at AR prices).
WRT the engine the 1.8 and 2.0L are both the same size physically, as in fact are the 1.4 and 1.6.
To check which it is you can either get the engine number off the block or open up the timing casing and have a look. If it's a 2.0L it will have a balance belt as well as timing belt..
btw if its registered as a 1.8 then it will be NCT emissions tested as a 1.8, but will give the results of a 2.0L. Not too sure if it matters but for various reasons, if it was me and it is a 2.0L engine I'd re-register it as that.