how do we demonstrate using Boolean algebra that these NAND and NOR gate combinations are XOR gates
The nice thing about small Boolean circuits like this, is there are only a few possible inputs.

With two input wires, there are $2^2=4$ possible inputs to your circuit: $00, 01, 10, 11$. You can test each one and write down its output. Then compare that to the XOR gate's outputs.

If the two line up, then the circuits are equivalent! And if they're not, something has to be adjusted.