Example 2 should work much better. The 3.3V regulator on the D1 Mini is the ME6211, which has a very low dropout voltage - nominally 100mV at 1A, but it's really somewhat higher than that. But the AMS1117 has a nominal 1.2V dropout voltage, meaning that its regulation will be compromised below 4.5V. Since your battery is at most 4.2V, the AMS1117 really isn't appropriate for single-cell lithiums. The ME6211 is just a better regulator.
Of course if you have a clone of the D1 Mini, the regulator might not be an ME6211.