彼は仰向けに寝転んだ。

English Translation

He lay on his back.

does 寝転ぶ have the connotation of “going from an upright to a laying down position,” or can it be used as “to already be in a laying down position” ?

I have found this definition for 仰向け:

顔や胸を上に向けた状態。- “The condition of having the face or torso facing up”

So it looks like it has the second meaning you listed.

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@ericaw @mike-lima
Your first interpretation is the right one. You were sitting on a couch or standing on the floor, and then lied down on the couch or bed. 転ぶ (ころぶ) means “to slip/tumble over”, so your body position is rotated 90 degree.

仰向け and うつ伏せ (うつぶせ) are face up and down, respectively.

If you are already lying on the couch/bed, and you just turn your body upside down, ソファー/ベッドの上で寝返りを打った is the expression. Pron. = うえでねがえりをうった.

ひっくり返す or 返す in short means “to turn something 180 degree” of such as a body, a steak on a grill or a trump card.