A seminal study by Libet et al. (1983) provided a popular approach to compare the introspective timing of movement execution (the M-time) and the intention to move (the W-time) with respect to the onset of the readiness potential (RP). The difference between the W-time and the RP onsets contributed significantly to the current free-will discussion, insofar as it has been repeatedly shown that the RP onset unequivocally precedes the W-time. However, the interpretations of Libet's paradigm continuously attract criticism, questioning the use of both the W-time and the RP onset as indicators of motor intention. In the current study, we further probe whether the W-time is rather an intention-unrelated product of the participant's inference than an unambiguous temporal marker of the intention to move. Using behavioral reports and concurrent multichannel EEG, we investigated the relationship between the W-time and M-time introspective reports in two groups of participants who started an experiment with a series of different reports. Congruently with previous studies, we have shown that the W-time is affected by the experimental procedures: participants who had prior experience reporting the M-time provided significantly earlier W-time. However, contrary to previous papers, we revealed that even naive participants do introspectively differentiate the W-time and the M-time, which suggests that the W-time might actually reflect the intention to move, at least to some extent. We, therefore, suggest that training-based modulation of the W-time values may explain this finding. Moreover, we further confirm the absence of a direct link between the RP onset and the W-time by showing no covariation between them in both experimental groups. In turn, our findings question the overall interpretation of the comparison between these two time points. Overall, our study further emphasizes the ambiguity of Libet's paradigm, and suggests that the relatedness of both the RP and the W-time to the movement initiation processes should not be assumed a priori.