Some things on car stand out like a sore thumb, big wheels, arches, bucket seats, modified dash, big boy exhaust, air filter, Carb (s) modified brakes and so on (4 pot calipers for example.) so these are pretty much straight forward if you go pre armed with a bit of info of what the standard car should have (Google or even ask on here)
Engine modifications can be a whole different story though so without a strip down its impossible to tell.
I hear all too often people saying don't bother telling them everything, well as an example, you crash into me with your Modified car (undeclared) and the Police get involved along with the insurance company. Now the first thing that happens is the Police (if involved) will look at the scene and both cars. They will issue out a form to me stating who they are, the drivers involved etc which can then be passed onto the the insurance company i will be claiming off (ie the person who crashed into me)
Now depending on the state of the vehicles they are towed away as they are no longer driveable. The insurance assessor will then visit these vehicles to see whether they can be repaired economically. He reports back and mentions that the car that caused the crash has what appears to be modifications and so the insurance company WILL check to see if they are declared.
They find they are not which then nulls the insurance. Myself will be informed i cannot claim against that policy and will have to claim against mine. This may or may not affect my NCB, etc etc etc. You the person who has caused the crash will have a damaged car and an insurance company chasing you for fraud and other things like damages etc etc. I will then chase you for expenses incurred through the courts if required.
Now certain people will say "this will never happen" good luck with that is all i will say. The grief i went through with the serious RTA i had over 5 years ago shows the lengths Insurance companies will go to.
I do know that age and cost plays a very significant part, been through all of that with my son and god help us when we finally get Project Erm on the road with whatever power plant he decides to install.
My 'Mini' is actually a kit car registered as an Austin Mini Minus. What was standard on a Mini Minus and what's a modification? Does it follow the mini that went into it, or is it what ever was fitted when it was built? Other than the details on the V5, it's impossible to say what is a modification and what went on when it was built.
Hopefully in your case a decent insurance company will recognise the car for what it is and what is fitted to it. I suspect you would have gone to a more specialised company.