Enquire and inquire

In British English I think these two words have different shades of meaning, but I couldn't articulate them. In American English I see inquire used where I would use "enquire".

Are there shades of meaning in British English?

Do they exist in American English?

The Oxford Learner's Dictionary has an "Help box" regarding Enquire vs Inquire, I'll paste it here:

In British English people sometimes distinguish between enquire and inquire, using enquire for the general meaning of ‘ask for information’ and inquire for the more particular meaning of ‘officially investigate’.

• "I called to enquire about train times."
• "A committee will inquire into the allegations."

However, you can use either spelling in either meaning. In American English inquire is usually used in both meanings.

The NOAD distinguishes them as inquire and inquiry being the usual U.S. spellings, while enquire and enquiry are classified as "the standard forms in Britain".

You almost always want enquire in the sense of "ask", eg "I enquired when the bus would leave"
Inquire is only used in the sense of an inquiry - an official investigation into something. eg. "Police inquire into the cause of the accident".

However my AE setting spellchecker doesn't think that "enquire" is a word - so this may be a British English distinction.