If it turns out that you're right (and I think what your proposed is the most likely scenario), then I would even go as far as saying it is more likely that early Mars seeded early Earth (as opposed to an Earth->Mars seeding). Although we don't know exactly what Mars was like 3.5-4 billion years ago, most evidence suggests that it was more temperate and stable than the early Earth.

In potentially the most interesting scenario, both Earth and Mars developed life independently, and Martian DNA-based life was carried to early Earth and completely eradicated whatever early non-DNA-based life had developed on Earth. It would perhaps explain why it is that all life on Earth is DNA-based. When Charles Darwin wrote "On the Origin of Species" it was not yet known whether all life was based on one basic code/template or whether there were multiple codes. Most scientists were surprised to find out that all life really did seem to have one DNA-based common ancestor. So either DNA-based life adapted a transformative mutation that enabled it to outcompete all other complex organic compounds, or DNA-based life was an extrinsic invader that took over the early Earth's biosphere.