Uhmm Dooby ... Nope , that's hopefully not the case.
While I bow to your experience in these matters ,in my very limited experience I kind of have to disagree . I have two motorbikes, both registered in Italy and both insured with two different insurance companies, neither of which actually issues a hard copy of my green card / insurance certificate anymore. I do however receive a PDF which I can print out if need to visit a country that's in the green card system but isn't in the EU, such as in Serbia where I am now and where I often visit. This is normally glanced ( and grunted) at on the border and gets no further mention.
I know that this is the same in Holland and some companies in Romania and probably elsewhere, there are a lot of people that have only a printout , like I do.
I do concede however that the OPs case is different to that of a European resident however I was not aware that not having a hard copy of your insurance was tantamount to not being insured at all, something too stupid to even risk doing .
I'm riding back from Novi Sad to Trieste tomorrow and I'll be sure to ask any Croatian police that I meet on the way, if I'm wrong , which wouldn't be the first time , then I'll take it as a welcome chance to learn something new. We can talk about it on Facebook if you like. Best greetings. Antonio.

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