Enquire means to 'seek information about' and inquire means 'to investigate'?
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Actually they both are the same words with the same meaning. The spelling 'enquire' is used in British Isles and commonwealth. The equivalent elsewhere is 'inquire'.

Hope this is clear now.
Been chasing this one down (again) today without finding any authority for the usage you suggest. Tried a series of dictionaries right back to a 1911 Chambers in our in-house library. The dictionaries all equate the words, defining 'inquire' and simply referring to that entry against 'enquire'.

My vote is very much with the 'nice' (in the sense of 'worth preserving') distinction you make. There is some support in practice. A coronial inquest is often referred to as a coronal inquiry, but never it seems as a coronial enquiry. Equally well, Committees of Parliament hold Inquiries about which they may well get enquiries. But they don't hold Enquiries.

I think your distinction is a good one to keep using. A more general questioning for information is well described as an enquiry. A more formal investigation resulting in a report is well described as an inquiry. I like the distinction, learned it from the nuns in primary school more than 50 years ago and choose to preserve it.

P.E. Dant