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We study the non-perfect propagation of information to evolving low-dimensional environment that includes self-evolution as well as noisy initial states and analyze interrelations between the degree of objectivization and environment parameters. In particular, we consider an analytical model of three interacting qubits and derive its objectivity parameters. The numerical analysis shows that the quality of the spectrum broadcast structure formed during the interaction may exhibit non-monotonicity both in the speed of self-dynamics of the environment as well as its mixedness. The former effect is particularly strong, showing that -- considering part of the environment as a measurement apparatus -- an increase of the external magnetic field acting on the environment may turn the very vague measurement into close to ideal. The above effects suggest that quantum objectivity may appear after increasing the dynamics of the environment, although not with respect to the pointer basis, but some other one which we call generalized pointer or indicator basis. Furthermore, it seems also that when the objectivity is poor it may be improved, at least by some amount, by increasing thermal noise. We provide further evidence of that by analyzing the upper bounds on distance to the set of states representing perfect objectivity in the case of a higher number of qubits.