It is not a misconception that increasing temperature increases the equilibrium (average) distance in almost all (atomic/molecular) systems. It is a generalization. It has exceptions.
Bruce! Temperature remains constant at primary phase transition!
Heat is a term which is not uniquely defined. It is the (net) flow of energy as used most often, although it also commonly refers to certain types of energy, such as 'that which is not work', or 'that which is available to do work'.
The relationship between distance and force between two unbonded atoms is a curve which is extremely repulsive (call that positive) at very close distances, decreases towards zero and then increases again to a secondary (local) maxima before declining monotonically as distance increases to ∞. The local minima is the equilibrium state (more or less). MOST materials (but not all) expand when heated. If I add heat to an explosive, do I increase its potential energy?
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I don't think many would prefer your phrase " [heat] separates the particles against their intermolecular forces, increasing their potential energy". It doesn't capture the meaning very well at all. Heat is not a force, so this sentence suggesting that energy acts "against" a force is awkward.
Energy input into a system (via heating) by definition either increases kinetic energy or increases potential energy (or both, in some systems the separation of energy into P.E. and K.E. is impossible) So, of course heat added to a system increases potential energy in the system (keeping velocities constant). It is the rest of the statement that is a problem. If you are familiar with a harmonic oscillator then you should understand that force is instantaneous, and that increasing the period increases the energy (and so the rms force) so while the force at certain instants is actually HIGHER than at lower frequency, the distance is also increased on average.