  We present Keck Cosmic Web Imager (KCWI) observations of giant Lya halos
surrounding 9 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 Lya luminosity and the
expected baryonic accretion rate (BAR), with increasing elongation above the
transition mass Mstream). This implies a modulation of the share of BAR that
remains cold diminishing quasi-linearly (logarithmic slope of 0.97+-0.19, 5
sigma significance) with the halo to Mstream mass ratio. The integrated
star-formation rates (SFRs) and AGN bolometric luminosities display a
potentially consistent decrease, albeit significant only at 2.6 sigma and 1.3
sigma, respectively. The higher scatter in these tracers suggests the Lya
emission might be mostly a direct product of cold accretion in these structures
rather than indirect, mediated by outflows and photo-ionization from SFR and
AGNs; this is also supported by energetics considerations. Below Mstream
(cold-stream regime) we measure LLya/BAR=10^{40.51+-0.16}~erg/s/Msun*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 Mstream (hot-accretion
regime), LLya is set by Mstream (within 0.2~dex scatter in our sample),
independent of the halo mass but rising tenfold from z=2 to 3.
