If momentum is conserved in a collision, does this indicate conclusively that the collision is elastic?

No, it doesn't. If a collision is elastic it means energy is conserved but it does not mean all the energy is conserved. If a collision is conclusively elastic it means there's is no loss of kinetic energy in the collision. Put in a different way: the total initial Kinetic Energy in the system must equal the total Kinetic Energy after the Collision. An elastic collision only shows that the collision is free of external forces; there's always a small loss due to heat and friction.