Position description for 'lying on the side propped up on one elbow with the hand supporting the head'

My try, being as succinct as possible using everyday, non-technical English, is

He was lying on his side, arm angled upwards, head on hand.

which seems to express the important points and make his position clear.

Another attempt is

He was lying on his side, his arm in a crook, head propped on his hand.

'Crook' here means 'a bend'. You could replace 'in a crook' with 'bent upwards', but my first suggestion seems better.

It seems important to mention his arm or elbow, because something such as "head supported on his hand" doesn't mean his hand was raised above the surface of the sofa.

I would describe it as “reclining”.

The Reclining Buddha is a widely occurring image of Buddhist semiotics. It is describe by the following features; “He is lying on his right side, his head resting on a cushion or relying on his right elbow, supporting his head with his hand.”

Source: Wikipedia

Reclining Buddha of the Hpo win caves: Link: Photo Source

Location: Burma

Completed: 14th Century

Link: Additional details regarding the Hpo Win caves

Nanzo-in

Founder: Hayashi Satoshiun

Completed: 1899

Location: 1035, Sasaguri, Sasaguri-machi, Kasuya-gun, Fukuoka, Japan

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River God (Arno)

“This statue, which shows the river god in a traditional reclining pose, dates from the time of the Emperor Hadrian and was inspired by a Greek prototype.”

Source: Vatican Museum

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Sarcophage d'un couple de défunt

“Portraits of the deceased In accordance with a practice that was also attested in Etruria, Italy, and Rome, the sculptor personalized this sarcophagus by portraying the deceased couple, half-reclining on the lid. The structure of the box resembles that of the funeral bed, supported at the corners by caryatids and terminal figures, with its thick mattress...”

Source: Astier Marie-Bénédicte , Louvre

Alas, it seems that there is no unique phrase that differentiates between the two images the OP provided