What do we call molten rock once it reaches the surface?

Magma below ground, lava above it - the same material, renamed at the moment it comes out. The distinction matters more than a naming convention usually would, because what changes at the surface is pressure, and pressure is what holds the dissolved gas in.

That is the whole story of whether a volcano oozes or explodes. Magma carries dissolved water and carbon dioxide the way a sealed bottle carries fizz. Release the pressure and the gas wants out. If the magma is runny it escapes easily and you get the rivers of lava that make Hawaii a tourist destination. If the magma is stiff and sticky the gas cannot get out until it shatters the rock around it, which is what Vesuvius and Mount St Helens did.

Stiffness comes down mostly to silica content and temperature. Basalt is low in silica, comes out at up to about 1,200 degrees Celsius and flows like syrup. Rhyolite is high in silica, cooler, and barely flows at all - it tends to pile up into a dome and then remove the top of the mountain.