The half-sari (pavadai davani in Tamil) is a “training” sari worn by adolescent girls in South India to ease the transition to the six yard sari worn by adult women. The three part ensemble (skirt, blouse, drape aka antariya, choli, uttariya) is not dissimilar to other 3 part outfits in India bar the fact that it mimics the “full sari” look.
Normally the lower part is a gathered skirt (like a ghaghra or pavadai worn by young girls) as in pics 2 and 3 but pic 1 (of the actress Savitri) is a little intriguing because the lower part has pleats like a sari but it is still a “half-sari”. I don’t think this is an earlier version of the dress (the photograph dates from the 50s), just a version worn in a movie.
The outfit wasn’t much worn in the recent past (the Northern salwar/chudidar ensembles are more common in South India of late) but it appears to be having a bit of a resurgence (for all ages, often with new materials and some hybridisation with the lehenga of the north) and as party/wedding wear.