The correspondent’s confusion is completely understandable: I suspect that nowadays most people would assume that a book described as a first edition implies, in effect, the first impression.
In the interest of absolute clarity, and to remove any lingering doubt in the average person’s mind, a professional bookseller could say First Edition, first impression.
If later impressions, without any (even minor) changes in format binding or dust jacket are issued, these should be indicated as ‘Second [or whichever] impression’, and described as such, without any mention of the words First Edition.
If there were any minor changes or corrections, it would become a variant issue, not a first impression.
I hope this helps.
Wilfrid de Freitas
PS While on the subject, please let’s stop using the expression true first edition; either it’s the first edition, or it isn’t. In the case of simultaneous UK and other countries’ (usually US) first editions, simply say First UK edition or First US edition and let the collector decide which she or he prefers to own.