Yes, for the lab demonstration and pictures that capture attention, the sample is kept cold enough to keep the water in ice form. The methane is trapped inside like shown below.
In this form, there is a lot more water than methane. The pilot projects attempted to find a economic production methods try to release the methane from the ice while still in place. It is far easier (cheaper) to handle gas in pipes like traditions natural gas production. Also far less material is moved and brought to the surface.
It doesn’t have to thaw to release the gas, just disturbing it, changing the pressure, can cause it to release “part” of the gas.
In warmer conditions, such as the ocean floor, it is true that sufficient depressurization will thaw the ice releasing the gas. Japan did that the second time, after warming the first time. Remember that steam, water, ice are functions of pressure as well as temperature. Both control the physical form of H2O
The only way "disturbing" it can sufficiently release the gas is if the material was already at the temperature/pressure line where it was ready to change state with little more energy.
BP's work in the arctic under the land on the North Slope required warming. It is too cold there for depressurization to work alone.