I've done it both ways in the past:
Going to your own insurer = Easier, less hassle, but you have a fault claim on your policy UNTIL they recover the money. So if your insurance renews in that period then it goes up (and then down again when settled).
Going to third party insurer = Sometimes more hassle but you don't have a fault claim on your own policy while the matter is settled.
Either way you should still inform your own insurer, as you have. Even a non-fault claim can affect your policy due to everything being prices using statistics (e.g. someone in your area may be statistically at risk of a second claim following a first).
Going to your own insurer = Easier, less hassle, but you have a fault claim on your policy UNTIL they recover the money. So if your insurance renews in that period then it goes up (and then down again when settled).
Going to third party insurer = Sometimes more hassle but you don't have a fault claim on your own policy while the matter is settled.
Either way you should still inform your own insurer, as you have. Even a non-fault claim can affect your policy due to everything being prices using statistics (e.g. someone in your area may be statistically at risk of a second claim following a first).
