This actually happened a few weeks back, but I've been on holiday for two weeks, so no chance to update before now.
After disputing the offer I was made by Churchill, I finally got to speak to a far more reasonable and understanding person in the Customer Complaints team. He agreed that it wasn't just about specification, but also about the brand of the product - even though this is not clear in the policy wording.
We haggled a bit over the final amount, but in the end I am pleased with the result and believe it was a good compromise that allows me to go out and buy a laptop of similar style.
It's a shame it took several complaints and emails before getting to that point - I even had to chase up after receiving a letter promising a prompt response, after I had waited two weeks.
Personally I think Churchill Insurance keeps the language on their policy deliberately vague in their policy for this sort of issue, as I doubt many people would push as hard as I did. If that is the case then I do not like that business style, as it comes across as trying to provide a minimum level of service and not caring about the quality. If you compare that with the level of service you get with a brand like Apple, it just doesn't stack up.
Anyway, a good result and yet another reminder that it pays to be persistent and stand up for your rights