Bodily illusions differently affect body representations underlying perception and action. We investigated whether this task dependence reflects two distinct dimensions of embodiment: the sense of agency and the sense of the body as a coherent whole. In experiment 1 the sense of agency was manipulated by comparing active versus passive movements during the induction phase in a video rubber hand illusion (vRHI) setup. After induction, proprioceptive biases were measured both by perceptual judgments of hand position, as well as by measuring end-point accuracy of subjects' active pointing movements to an external object with the affected hand. The results showed, first, that the vRHI is largely perceptual: passive perceptual localisation judgments were altered, but end-point accuracy of active pointing responses with the affected hand to an external object was unaffected. Second, within the perceptual judgments, there was a novel congruence effect, such that perceptual biases were larger following passive induction of vRHI than following active induction. There was a trend for the converse effect for pointing responses, with larger pointing bias following active induction. In experiment 2, we used the traditional RHI to investigate the coherence of body representation by synchronous stimulation of either matching or mismatching fingers on the rubber hand and the participant's own hand. Stimulation of matching fingers induced a local proprioceptive bias for only the stimulated finger, but did not affect the perceived shape of the hand as a whole. In contrast, stimulation of spatially mismatching fingers eliminated the RHI entirely. The present results show that (i) the sense of agency during illusion induction has specific effects, depending on whether we represent our body for perception or to guide action, and (ii) representations of specific body parts can be altered without affecting perception of the spatial configuration of the body as a whole.