The bottom LED in the right hand circuit is upside-down. If you correct this this, then the right hand circuit is the correct approach.
Drawing the left-hand design in the two conditions, output low (below left) and output high (below right), we can see why it's less efficient, and won't work for 3.3V logic and 1.8V LEDs:

simulate this circuit – Schematic created using CircuitLab
If the diodes were absent, the potential at X would be half the supply potential, leaving R16 and R17 with 1.65V across each, which is insufficient to light either LED. Installing an LED would only serve to reduce this already insufficient potential difference even further. The system could work with a supply of 4V or higher, but regardless of output state, there's always a resistor in parallel with the diodes dissipating power unnecessarily.
Drawing your right-hand design in the two possible states, we have a very different situation, illustrating how those two designs are far from equivalent:

simulate this circuit
In this arrangement, one of the LED+resistor pairs has the full supply voltage across it, regardless of the digital output state, or power supply. The "inactive" resistor and LED have 0V across them, and dissipate no power.
For a more rigorous appraisal of your left hand design, replace the system of R16, R17 and a 5V supply, with their Thevenin equivalent:

simulate this circuit
If we were to calculate an appropriate resistance Rth to pass diode current \$I_{D}=1mA\$ through D, when D has voltage \$V_{D}=1.8V\$ across it:
$$
\begin{aligned}
R_{TH} &= \frac{V_{TH}-V_D}{I_D} \\ \\
&= \frac{+2.5V - 1.8V}{1mA} \\ \\
&= 700\Omega \\ \\
\end{aligned}
$$
This corresponds to two resistances \$R_{16}=R_{17}=R=2R_{TH}=1.4k\Omega\$.
The same 1mA of diode current in your right hand circuit would use resistances of
$$R_{18}=R_{19}=\frac{5V-1.8V}{1mA}=3.2k\Omega$$
Now let's plug our value of 1.4kΩ for R16 and R17 back into the your left circuit, and take a look at the various resulting currents and voltages:

simulate this circuit
We have the required 1.8V across, and 1mA through D1, but there is an additional current of 1.3mA through R16. Clearly that's wasting power, more power in fact, than the LED is receiving!