If we found evidence of life on Mars, how would we know that it originated on Mars rather than Earth?

Why would we assume that the early martian life originated on Mars,
rather than Earth?

There's still a whole lot we don't know. As Wayfaring stranger points out in the comments, Origin is a whole different question. It's possible that life originated outside our solar system and came to either Mars and/or Earth from outside the solar-system. I don't think anyone who studies this idea is "Assuming" life originated on Mars, only that the idea has a chance of being true.

Is it not likely that somewhere between 2.5-3 billion years ago, a
major volcanic eruption or meteor impact could have hit an earth
teeming with simple-life and sent it hurdling through space to Mars?

As I understand it, volcanic eruptions are unlikely to send anything into space unless it's a smaller sized moon. The escape velocity (young earth, maybe 10 km/s, young mars, maybe 3-4), volcanic eruptions, as far as I know, don't shoot things out at 10,000 - 20,000 MPH. But meteor impacts of sufficient size can do that.

Mars is a better meteor debris making target than the Earth cause it's smaller, so the gravity is lower and presumably it's mostly had a thinner atmosphere too. We've found martial meteors on Earth. We might not find any Earth meteors on mars cause it takes a much bigger impact to knock bits of rock off Earth and because the Atmosphere slows objects down, both coming in and going out.

I understand that my theory has the gravity of the sun working against
it, but I also see it not being entirely impossible.

The sun isn't as big a factor as you might think. Once something is knocked off a planet and it gets into orbit around the sun, gravitational assists can move it around further out or further inside the solar system. What presumably happens is that, with a big enough impact, many thousands if not millions of bits of debris get into solar-system orbit and from there, some of them land on other planets - probably much less than 1% of those hit Earth, but if it carries life that can survive the trip, all you need is one rock.

I mean, 100 years ago, one could assume that life originating on Mars
and then being sent this was was ridiculous and impossible but now
it's a rational, viable theory.

While that's true, the "People used to think this was impossible" isn't a scientific approach for what might be true. We should determine what is possible and/or thought to be likely, based on physical evidence, not what wasn't understood 100 years ago. Your example is a good point on why it's important to keep an open mind about the unknown. You can still make theories based on evidence, and keep an open mind on the unknown. There's really no conflict between the two.

Considering such a scenario, how would scientist verify that these
life-forms didn't originate from Earth, rather than Mars? It seems to
me that this question is the first question that would need to be
asked and answered after the fossil-records were found and verified.

It's a good question.

The simple answer is that Mars cooled first and Mars (likely) had oceans first, so it's a better candidate to have developed life first though extremophiles can live in hot oceans, so . . . time will tell.