I think a great deal also depends on where you physically are located - in which or whose country this mythical introduction is being placed.
If I remember correctly, Charles of Wales came to the United States on a visit in the early 1980's. I'd say 1981, but I'll have to look that up. As part of his visit, he flew to California to stay at Sunnylands, the estate of the former US Ambassador to the UK, Walter Annenberg. Annenberg was quite popular during his tenure in the UK, and received an OBE. While in the UK, he and his wife could be known as Sir and Lady Annenberg, which of course had no such effect at home in the US.
When Charles came down the steps of his airplane, he was greeted at the bottom of the stairs by Mrs. Annenberg, who practically fell over herself to throw herself at Charles' feet. In California. Embarrassing for all, really. The papers had quite a field day, printing and reprinting the picture of the wife of the former US Ambassador hurling herself groundward in a curtsey. And reason for the furore was this: in the United States, on US soil, curtseying and bowing are simply not done. Not. Done. Period. We are a Republic, and that is not done.
Mrs. Annenberg bleated that she always curtsied to Charles and the RF when they were visiting England, and the response is yes: that's right. Their country, their customs, their rank, bow/courtesy all you like. This was US soil; no bowing, no curtseying, no titles: stop pretending you're somewhere or someone that you are not.
Back to the recent visit of the French president and his wife. As the wife of a head of state, Carla Bruni is never required to curtsey to a foreign sovereign. That she chose to do so was charming.
Bunny Mellon was the unofficial advisor to Jackie Kennedy during the Kennedy administration, and advised her that, as the wife of a head of state, she did not have to bow to foreign royalty on their soil, although it otherwise might be required.
If I were visiting the King and Queen of Sweden in their summer home - and I need to check the mail tonight to see if my invitation is there or if it was mislaid - it would be my obligation to them as their guest in their country of which they are sovereign to pay appropriate courtesy. If I invited them to our house on the Finger Lakes region of upstate New York for a summer visit, I would not expect to curtsey to them.
I do need to check the mail.