slight correction to Random's substitutions (community wiki since it's not my contribution):

define: $${n_0}=n+x+y-k,\;\;{i_0}=i+k-y-z,\;\;{j_0}=j+k-x+z$$ substitute in Vajda's identity $$F_{n_0+i_0}F_{n_0+j_0}-F_{n_0}F_{n_0+i_0+j_0}=(-1)^{n_0}F_{i_0}F_{j_0}$$ and you obtain the first equation $$F_{n+i+x-z}F_{n+j+y+z}-F_{n+x+y-k}F_{n+i+j+k}=(-1)^{n+x+y-k}F_{i+k-y-z}F_{j+k-x+z}.$$

Slight correction to Random's substitutions (community wiki since it's not my contribution):

define: $${n_0}=n+x+y-k,\;\;{i_0}=i+k-y-z,\;\;{j_0}=j+k-x+z$$ substitute in Vajda's identity $$F_{n_0+i_0}F_{n_0+j_0}-F_{n_0}F_{n_0+i_0+j_0}=(-1)^{n_0}F_{i_0}F_{j_0}$$ and you obtain the first equation in the OP, $$F_{n+i+x-z}F_{n+j+y+z}-F_{n+x+y-k}F_{n+i+j+k}=(-1)^{n+x+y-k}F_{i+k-y-z}F_{j+k-x+z}.$$ So there is no generalization involved.