If the Sun were smaller but had the same surface temperature, would it still have the same luminance? Let's assume we have two stars that have the same surface temperature but very different size. I understand how luminosity depends on surface area so the two stars will have different luminosity, but how about luminance?
In other words, would it be correct to say that the luminance of a star depends only on its surface temperature so both of the stars will have the same luminance? Does the luminance depend on the surface area so the two stars will have different luminance?
 A: Both From Wikipedia Luminance Versus Luminosity

In astronomy, luminosity is the total amount of energy emitted by a star, galaxy, or other astronomical object per unit time. It is related to the brightness, which is the luminosity of an object in a given spectral region.

Now your question:

Let's assume we have two stars that have the same surface temperature but very different size. I understand how luminosity depends on surface area so the two stars will have different luminosity. But how about luminance ? In other words, would it be correct to say that the luminance of a star depends only on its surface temperature so both of the stars will have the same luminance ? Or the luminance depends on the surface area so the two stars will have different luminance ?

Wikipedia again:

Luminance is a photometric measure of the luminous intensity per unit area of light travelling in a given direction. It describes the amount of light that passes through, is emitted or reflected from a particular area, and falls within a given solid angle. The SI unit for luminance is candela per square metre (cd/m2). In geometry, a solid angle (symbol: Ω) is the two-dimensional angle in three-dimensional space that an object subtends at a point. It is a measure of how large the object appears to an observer looking from that point. In the International System of Units (SI), a solid angle is expressed in a dimensionless unit called a steradian (symbol: sr).
A small object nearby may subtend the same solid angle as a larger object farther away. For example, although the Moon is much smaller than the Sun, it is also much closer to Earth. Indeed, as viewed from any point on Earth, both objects have approximately the same solid angle as well as apparent size. This is evident during a solar eclipse.

In other words, taking Luminosity first, that is a measure on the total power emitted by the star, independent of distance from it, whereas Luminance as described above, is simply a measure of how much light you are receiving, and may not necessarily come directly from the star, but be reflected from the moon, say, or diffused through interstellar gas.
So luminance is not directly associated with the surface temperature of the star, and is independent of the surface area of the original light source, although a much more direct connection can be made for luminosity, as given in the definition above.
Just to finish off, From Wikipedia Solid Angle

The solid angle of a sphere measured from a point in its interior is 4π sr, and the solid angle subtended at the center of a cube by one of its faces is one-sixth of that, or 2π/3 sr. Solid angles can also be measured in square degrees (1 sr = (180/π)2 square degree) or in fractions of the sphere (i.e., fractional area), 1 sr = 1/4π fractional area.

Sr above means steradian. Any area on a sphere which is equal in area to the square of its radius, when observed from its center, subtends precisely one steradian.
This illustration may help in visualising how the original light source spreads out from a central point.  In crude terms, the central point, say a star, has a certain luminosity, wheras the luminance is the light you measure passing through the red section.
As I say above, the emitted light will, in all probability, have ben reflected and diffused by the time you measure it, compared to the possibly misleading "clear run" appearance it is displayed as having in the illustration.
EDIT In physics, intensity is the power transferred per unit area, which is transmitted through an imagined surface perpendicular to the propagation direction.1 In the SI system, it has units watts per square metre (W/m2). It is used most frequently with waves (e.g. sound or light), in which case the average power transfer over one period of the wave is used. Intensity can be applied to other circumstances where energy is transferred. For example, one could calculate the intensity of the kinetic energy carried by drops of water from a garden sprinkler.
The word "intensity" as used here is not synonymous with "strength", "amplitude", "magnitude", or "level", as it sometimes is in colloquial speech.
Intensity can be found by taking the energy density (energy per unit volume) at a point in space and multiplying it by the velocity at which the energy is moving. The resulting vector has the units of power divided by area. END EDIT

