I think you just need to watch a few details of the terminology - binary, rounded binary, ternary, small ternary, large ternary - and understand rounded binary and small ternary are synonymous. "Ternary" is a bit too generic for your question.
Superficially A B A
applies to both rounded binary and large ternary forms.
Rounded binary uses those letters but in this specific kind of structure: ||: A :||: B A' :||
.
Large binary uses those letters but the idea is the A
and B
sections are distinct and self contained sections.
The difference is mostly about the stuff labelled B
. In rounded binary B
is continuation and contrast material, but in large binary is has a clear expository function.
An important concept related to all this is standing on the dominant which is characteristic of the B
of rounded binary. Standing on the dominant is not expository. It's harmonic contrast and continuation. It prepares for a return to the opening tonic.
In this song the stuff to label B
is mostly about standing on the dominant, it's not expository. It doesn't present a new key, it's not a new theme.
This song would be called either rounded binary or small ternary form.
William Caplin's Classical Form discusses all these concepts in great detail.
A form that demonstrates both types of A B A
in one composition is the menuet and trio. An example is Haydn's Haydn Sonata in C Major, Hob XVI 1. The menuet is the A
part, the trio is the B
part, and the second A
part is the da capo repeat of the menuet. That is a larger ternary form. But, the menuet and trio are both examples of rounded binary (or small ternary) form ||: A :||: B A' :||
.
19th century pieces like piano dances or preludes also provide good examples of large ternary form. Look at examples like Chopin's Mazurkas - Mazurka Op. 24 No. 2 and Op. 17 No. 4 - are nice examples with double barlines and key/mode changes marking out the B
sections clearly.