Across resistor +/- 1.32V
Across LM317 +/- 0.6V
You are going to tell me that LM has a dropout of more than 2V... and you are right.
But that dropout is for full load 1 to 1,5A.
At 60mA and @ 25ºC the dropout is +/- 1,6V...
So, The circuit must be corrected
Assuming a charge voltage of 6,9V and a current of 60mA:
Vbat + Ve1 + Vdropout + (Vdiode x 4) =
6,9 + 1,32 + 1,6 + (0,6 x 4) =
6,9 + 1,32 + 1,6 + 2,4 =
12,22V
To get the transformer voltage, we must convert this into RMS value:
12,22V / 1,414 = 8,64V
Finally, a transformer from 8V to 9V is sufficient
But... the first schematic also works, but not as efficient