Regarding the title: Yes.
Does this mean that the star started off as a planet?
Yes, a star could technically start out as a planet, if it accreted enough mass. However, this is extremely unlikely, since the planet would need to be 80x the mass of Jupiter for it to undergo nucleosynthesis.
Stars require hydrogen fusion and earth has little H. Could the earth become a star if more mass was added, but its relatively low abundance of hydrogen remained the same?
The most important components for a star is the composition and mass. As it stands, Earth doesn't have nearly the concentration of hydrogen for a star to form, even if we kept adding mass.
Most stars work by fusing the lightest element, hydrogen. To fuse heavier elements requires a much, much higher temperature, since the more protons the element has, the more it will repel other ones instead of fusing. Yes, massive enough stars can fuse heavier elements.
However, Earth would inevitably grow in hydrogen, so it would eventually have a significant amount. Above a certain size, given ambient conditions (like radiation pressure, light intensity from the star, etc), Earth would become massive enough that it starts capturing hydrogen gas from the interplanetary medium, and becomes a gas giant (as called2voyage pointed out). Where this happens isn't precisely known, but it's roughly 10x Earth's mass. This means that, theoretically, a terrestrial planet could form a star if you increased its mass ever further.
Now, I can hear you thinking, "What if Earth were surrounded by a vacuum instead? If we kept adding mass, would it become a star?" Well theoretically, the planet would eventually reach temperatures where most of its elements could fuse. For example, the "star" could sustain itself on oxygen and silicon fusion, but not on iron fusion.