High HC (Hydrocarbon) readings in the coolant system means exhaust gasses and or engine oil are getting in somehow, and the thing that normally prevents that is the head gasket, add in the overheating and I'd say it's a very strong indicator, not perhaps definitive proof.
That said, since most of most of the other options I can think of for how that can happen are worse than a blown head gasket I think, as strange as it might sound, you might have to hope that it's only the head gasket that's failed.
We don't really do shopping advice here (and I know you didn't explicitly ask) but the quoted price sounds a bit on the high side for just a head gasket replacement to me, but it depends on whether they're looking at replacing other bits (the i20 uses a timing chain so the normal "replacing the timing belt at the same time" doesn't apply here but it's safe to say a full change of oil and coolant would be sensible) or whether they're building in some headroom in case they need to skim the head due to warping.