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A thorough study and analysis on the conceptual foundations of unimodular gravity shows that this theory is essentially general relativity disguised as unimodular relativity in the literature. The main reason for this dilemma is accepting the Einsteinian paradigm: general relativistic framework, covariant divergence law for matter energy-momentum tensor, and cosmological constant as an integration constant, but introducing the artefact of unimodular description absent in Einstein's work. A new approach is proposed in this paper in which pure unimodular relativity is defined in terms of equi-projective geodesics with the fundamental metric tensor having determinant unity and the geometric tensors constructed from them. Modification of covariant divergence law for the matter energy-momentum tensor is shown to have two new consequences. In the conventional unimodular gravity an effective cosmological term comprising of two variable scalar fields, namely, the unimodular geometric ambiguity and unimodular matter energy ambiguity, is proposed. A radical departure on the cosmological constant problem is possible assuming differing evolution of the two scalars: the Einstein equations emerge when the two ambiguities cancel each other. Secondly, in the case of pure unimodular relativity the gravitational field equations are proposed consistent with the unimodular space-time structure.