ca. 1780 French silk and metallic thread caraco (Metropolitan Museum of Art - New York City, New York USA) front, side, and detail

Americanduchess takes a shot at defining all of these jacket styles - after a cautionary note:

"Disclaimer:  The information in this post is what I have gleened from sources, but the lines seem very blurry between these styles, and so if I am wrong about something, please leave me a comment and correct me!

A caraco is a long-length jacket with a fitted back, like a robe a l'Anglaise. The length of the skirt of these jackets seems to be about mid-thigh, and the skirts usually have inverted box pleats at the back, often pressed but sometimes left loose. Caracos typically have 3/4 length sleeves with flounces.  Examples of caracos close severals ways, with lacings over a stomacher, pinned to a stomacher, with "flaps" that hook across a stomacher, or with a comperes front (a false front made to look like a stomacher, that closes at the center with hooks or buttons), or edge-to-edge by hooks.

A pet en l'ier jacket is a garment of mid-thigh to mid-hip length, with a saque back, that is, watteau pleats at the back.  This jacket usually has 3/4 sleeves with flounces, and closes over a stomacher or with a comperes front. Now get ready for this - "pet en l'ier" typed into Google translate comes out to "fart in the street."  If this is wrong, please correct me, my faithful French readers!!  

A pierrot is a a very short jacket, actually more of a bodice with a ruffle or flounce added onto the back.  Pierrot jackets came into style later in the century, and often closed edge-to-edge with hooks or interior lacings.  They were most commonly long-sleeved, and worn with walking-length skirts, puffy fichus, and monster-sized hats. "Pierrot," in English, means "sparrow." 

A casaquin jacket is that of short length, about mid-hip, but still with a flared skirt and pleats at the back.  It has a fitted back, like a robe a l'anglaise, and may or may not have seams at the waist.  Casaquins may have 3/4 length of full-length sleeves, and close over a stomacher.  The hallmark of a casaquin appears to be the lacey trimmings."

This caraco at the Metropolitan Museum from around 1780 has back pleats, meeting Alewyn's and americanduchess' definitions and it is also long. This example also appears to have a comperes front.

Keywords:  1780, French, trapezoidal neckline, caraco, modesty piece, comperes front, buttons, panniers
