As you say, the decibel is a unit of power ratio.

\$ G\ [\mathrm{dB}] = 10 \log_{10}(\dfrac{P_1}{P_2})\$.

When the input and output impedances are equal and then we can express the gain in terms of voltage as

\$ G\ [\mathrm{dB}] = 20 \log_{10}(\dfrac{V_1}{V_2}) \$

I wouldn't call this the "voltage gain in decibels." I'd rather say it's the decibel gain, calculated from the voltage gain.

Sometimes, you will see a voltage gain expressed in decibel according to this formula even when the input and output impedances are different. There is no technical justification for this --- it's simply a shorthand practice that's become common through usage.

As you say, the decibel is a unit of power ratio.

\$ G\ [\mathrm{dB}] = 10 \log_{10}\left(\dfrac{P_1}{P_2}\right)\$.

When the input and output impedances are equal and then we can express the gain in terms of voltage as

\$ G\ [\mathrm{dB}] = 20 \log_{10}\left(\dfrac{V_1}{V_2}\right) \$

I wouldn't call this the "voltage gain in decibels." I'd rather say it's the decibel gain, calculated from the voltage gain.

Sometimes, you will see a voltage gain expressed in decibel according to this formula even when the input and output impedances are different. There is no technical justification for this --- it's simply a shorthand practice that's become common through usage.