Eww, the connectors aren't on grid. Hmm, B's ground is floating, too.
The phasing of B is... suspect, but maybe not out-and-out wrong.
Well, assuming those are fixed in the most obvious manner:
These are a simple transformation (ha) of a circuit, with a simple rule: voltage-mode vs. current-mode.
The reason cannot be understood with an ideal transformer: in that case, there is no difference, no advantage to one or the other.
Understanding the nonideal transformer brings relevant insight:

In A, stray capacitance (or even more nuanced: transmission line effects) affects balance and bandwidth, setting an upper cutoff; magnetizing inductance (and coupling capacitance where applicable) sets a lower cutoff.
In B, magnetizing inductance affects balance, but bandwidth is essentially unlimited (the two windings can be a single transmission line). But I think a mistake has been made here:
Mentally flip around the bottom winding (pins 1/2), so that the bottom output (nets aren't labeled, so I'll use C4/C9 to identify them instead; so, C9) comes from the left side and the dot is on the right. Now we have a common-mode choke, going between input+C9, through a transformer core, to C4+GND. If J3 has a rising step applied, C9 voltage rises immediately, given by the divider of (transmission line) Zo into C9 termination resistance (let's say it's R to GND). If Zo = R, this is 50%, after a delay of ~zero. One electrical length of time later (the propagation delay of the TL), C4 rises to half, and reflects back its load (let's say R again).
It's a 2:1 network (say 100Ω input, 50+50Ω output), but a balun it ain't. It's a power combiner.
If T5 windings have dots on the same side, it's a CMC and a "current balun" in the usual way (follow the same analysis as above).
Tim