The story of Dido's suicide existed 1700 years before Purcell came along. It's in Vergil's Aeneid, book 4. Here's the relevant bit, where she stabs herself with a sword, while making speeches.

The motif of a 'lament', as such, probably owes more to Purcell's famous aria, but one of her speeches in Vergil is a lament too; and it's probably worth observing that the painters you point to are depicting her death, not a lament specifically.

Vergil is the ultimate source for the death narrative, but Dido herself was a traditional character in his time too. In earlier stories she's called Elissa, and she had nothing to do with Aeneas. Here's an answer from last year by me that talks a bit about Vergil's sources; the last two paragraphs touch on Dido/Elissa.

I love this sub. Learning so much every day. Thank you for posting questions and those for answering.