You are right that the term "burn more" is too vague. It could refer to the extent of the burn, how fast the burn occurs, etc.

In any case, the minimum necessary (but not sufficient) condition for a skin burn is the temperature of the object must exceed a certain threshold. Then the severity of a skin burn is determined by the combination of the rate of heat transfer to the skin and the duration of the exposure (contact time). The rate of heat transfer depends on the thermal conductivity and temperature of the object. The product of the heat transfer rate and exposure time is the amount of heat transferred. This is where the heat capacity of the object comes into play.

Generally, the heat capacity of the object only becomes a factor when dealing with contact with small, or low mass parts. A familiar example is aluminum foil. It is possible to touch aluminum foil taken directly from the oven without getting burned, even though the temperature and the thermal conductivity of the foil is very high. This is because the foil is very thin. The product of the contact area with the foil (about a square centimeter for a finger pad) and the thickness of the foil (about 0.016 mm) and its specific heat (about 0.9 kJ/kg C) results in a very low heat capacity, so low that the amount of heat available from the foil is usually insufficient to cause a skin burn.

For more details about the various factors that go into producing a skin burn, see my answer to the following:

Why does holding a hot object with a cloth make it feel less hot?

Hope this helps.

You are right that the term "burn more" is too vague. It could refer to the extent of the burn, how fast the burn occurs, etc.

In any case, the minimum necessary (but not sufficient) condition for a skin burn is the temperature of the object must exceed a certain threshold. Then the severity of a skin burn is determined by the combination of the rate of heat transfer to the skin and the duration of the exposure (contact time). The rate of heat transfer depends on the thermal conductivity and temperature of the object. The product of the heat transfer rate and exposure time is the amount of heat transferred. This is where the heat capacity of the object comes into play.

Generally, the heat capacity of the object only becomes a factor when dealing with contact with small, or low mass parts. A familiar example is aluminum foil. It is possible to touch aluminum foil taken directly from the oven without getting burned, even though the temperature and the thermal conductivity of the foil is very high. This is because the foil is very thin. The combination of the contact area with the foil (about a square centimeter for a finger pad), the thickness of the foil (about 0.016 mm) its density (about 2.7 g/cm$^3$) and its specific heat (about 0.9 kJ/kg C) results in a very low heat capacity, so low that the amount of heat available from the foil is insufficient to cause a skin burn.

For more details about the various factors that go into producing a skin burn, see my answer to the following:

Why does holding a hot object with a cloth make it feel less hot?

Hope this helps.