We present Keck Cosmic Web Imager observations of giant Lyα halos surrounding nine galaxy groups and clusters at 2 < z < 3.3, including five new detections and one upper limit. We find observational evidence for the cold-stream to hot-accretion transition predicted by theory by measuring a decrease in the ratio between the spatially extended Lyα luminosity and the expected baryonic accretion rate (BAR), with increasing elongation above the transition mass (M stream). This implies a modulation of the share of BAR that remains cold, diminishing quasi-linearly (logarithmic slope of 0.97 ± 0.19, 5σ significance) with the halo to M stream mass ratio. The integrated star formation rates (SFRs) and active galactic nucleus (AGN) bolometric luminosities display a potentially consistent decrease, albeit significant only at 2.6σ and 1.3σ, respectively. The higher scatter in these tracers suggests the Lyα emission might be mostly a direct product of cold accretion in these structures rather than indirect, mediated by outflows and photoionization from SFR and AGNs; this is also supported by energetics considerations. Below M stream (cold-stream regime), we measure L Lyα /BAR = 1040.51±0.16 erg s-1 M-1 yr, consistent with predictions, and SFR/BAR = 10-0.54±0.23: on average, 30-10+20 % of the cold streams go into stars. Above M stream (hot-accretion regime), L Lyα is set by M stream (within 0.2 dex scatter in our sample), independent of the halo mass but rising 10-fold from z = 2 to 3.