Which is more amphoteric: zinc or copper hydroxide?
Zinc hydroxide is more amphoteric as quantitatively investigated in Effect of pH, Concentration and Temperature on Copper and
Zinc Hydroxide Formation/Precipitation in Solution 

In summary, in the case of Cu(OH)2, only at extremely low concentrations and very high pH can a third OH- be coordinated.


  At higher pH values, copper
  hydroxide, Cu(OH)2, is the dominant species up to pH 12.3 where the copper ion
  Cu(OH)3
  - forms according to Equation 3. At higher copper concentrations, solid
  Cu(OH)2 is formed and precipitates out of solution at copper concentrations above the
  solubility product of copper hydroxide at 1×10-8 M. It is important to note that the
  domain of stability of solid Cu(OH)2 is expanding to lower and higher pH values with
  increasing copper concentration.


On the other hand:


  Above pH 11.4, Zn(OH)3
  - forms according to Equation 7. At
  higher zinc concentrations, solid Zn(OH)2 forms and precipitates out of solution at zinc
  concentrations above the solubility product of zinc hydroxide of 1×10-5 M.


So qualitatively both display similar amphoteric behavior in the limit of infinite dilution, but at a appreciably non-zero concentration zinc is more amphoteric.    

The article says it is getting the equilibrium constant values from Mineral Equilibria, Low Temperature and Pressure.  

An alternative source of this information is Hydrolysis of cations. Formation constants and standard free energies of formation of hydroxy complexes Inorg. Chem., 1983, 22 (16), pp 2297–2305, which gives quantitative values for OP equations 1, 2 and 3 as well as the corresponding equations for zinc, in terms of Gibbs energy of formation and in terms of cumulative equilibrium constants.

Simplifying the general equation in the paper to only mononuclear (one metal atom) complexes: 

$$K_y$$
$$\ce{M^{n+} + yH2O &lt;=&gt; M(OH)_y^{n-y} + yH+}$$

$\ce{Cu^{2+}}$ :
$pK_1 = 7.96$
$pK_2 = 16.26$
$pK_3 = 26.7$
$pK_4 = 39.6$  

$\ce{Zn^{2+}}$ :
$pK_1 = 8.96$
$pK_2 = 16.9$
$pK_3 = 28.4$
$pK_4 = 41.2$  

(These are all experimental values, calculated values are also given)

So really, without considering the solid phase (solubility), Cu2+ and Zn2+ seem very similar, with Cu2+ showing slightly more acidity.  

Another study, which does consider the solid phases, is Zinc Hydroxide: Solubility Product and Hydroxy-complex Stability
Constants from 12.5-75 [degrees] C Can. J. Chem. 53, 3841.  

This study approaches the problem with 5 equilibria involving the solid phase. (These authors use the symbol "c" for the solid phase.)  

$\ce{Zn(OH)2(c) &lt;=&gt; Zn(OH)+ + OH-}$
$K_1 = [\ce{Zn(OH)+}][\ce{OH-}] = 2.54 \times 10^{-11}$ 

$\ce{Zn(OH)2(c) &lt;=&gt; Zn(OH)2(aq)}$
$K_2 = [\ce{Zn(OH)2}] = 2.62 \times 10^{-6}$  

$\ce{Zn(OH)2(c) +OH- &lt;=&gt; Zn(OH)3-}$
$K_3 = [\ce{Zn(OH)3-}]/[\ce{OH-}] = 1.32 \times 10^{-3}$  

$\ce{Zn(OH)2(c) +2OH- &lt;=&gt; Zn(OH)4^{2-}}$
$K_4 = [\ce{Zn(OH)4^{2-}}]/[\ce{OH-}]^2 = 6.47 \times 10^{-2}$

$\ce{Zn(OH)2(c) &lt;=&gt; Zn^{2+}} + \ce{2OH-}$
$K_{sp} = [\ce{Zn^{2+}}][\ce{OH-}]^2 = 1.74 \times 10^{-17}$

This study (see Fig. 1) shows that aqueous $\ce{Zn(OH)2}$ is only the major aqueous species in the pH range 9-11, having acted as an acid or base outside this range.  



Instead, if I were just given the information in the OP, meaning the two supposed Ka values, as some kind of test or homework question,  I would say copper is more amphoteric.  My reasoning would be, for zinc, just to remove one proton, you need to get up to almost pH 10.  For two more protons to be removed, as required to get to Zn(OH)2 and lose yet another proton to act as an acid, seems impossible because I would expect pKas to be spaced apart and if you would need to approach or go beyond pH 14 to observe the acidic behavior, that isn't reasonably considered amphoteric.  

Note that EXAFS Investigations of Zn(II) in Concentrated Aqueous Hydroxide Solutions J. Phys. Chem. 1995, 99, 11967-11973 finds that rather than being 6-coordinate as indicated in the OP, 4 hydroxides coordinate Zn2+ tetrahedrally with no water ligands.