RV9 pilot reported they misunderstood ATC traffic instruction resulted in an NMAC on final.

Narrative

I was doing touch-and-goes; making left closed traffic; ZZZ Runway XXL; in my RV-9A. I was turning crosswind-to-downwind when Tower radioed*; 'Aircraft X number two following a light-sport on left base to a three-mile final Runway XX left cleared for the option.' I read back the clearance and completed the downwind turn; looking from my 12 to 2 o'clock for the traffic on left base to XXL. Visibility was 5sm with haze; which was backlit by the low winter sun. It was effectively impossible to see any traffic west of the airport. However; I did _not_ report negative visual contact to Tower in my read-back.Proceeding on the downwind at about midfield I noted a low-wing aircraft just lifting off from XXL. Two nearly simultaneous thoughts popped up in my head: 1) That could be the light-sport traffic; 2) How could a light-sport get from base to a 3 mile final to touchdown so fast? I didn't conclude it was the light-sport. But I also didn't ask Tower to confirm one way or the other.Abeam the numbers; I had still seen no traffic. It being common practice for ZZZ Tower to instruct ZZZ pilots to extend their downwind legs when conflicting traffic is on final; and having received no such instruction; I concluded as I prepared to turn base I must be number one for XXL by now. Again; in retrospect; I should have confirmed with Tower to be sure. Still seeing no traffic on final as I turned base; and after turning final there was nothing but empty air between me and the numbers. Then Tower advised; 'Aircraft X; your traffic to follow; light sport; uh; half mile final.' What? But there's nothing out there. And _I'm_ already about half a mile from the threshold!?I searched the windshield again for a few seconds; then replied; 'Aircraft X doesn't have that traff--uh; oh my God; yes--.' A white high-wing airplane on my heading emerged from directly underneath my nose. Vertical separation was about 200 feet; perhaps less.Tower immediately radioed; 'Aircraft X; go around;' but I don't recall hearing it. I was already powering up and offsetting my flight path to the left over Taxiway 1. I radioed; 'Aircraft X will climb and go around;' shortly thereafter.Tower then advised the light-sport that they were number one for Runway XXL and that 'traffic in trail is going around.' Shortly thereafter; Tower radioed; 'Aircraft X; early turnout approved if you want it. Runway XX left; cleared for the option you're number one.'I started to turn cross-wind; replied; 'Cleared for the option; uh; number one; this will be a full stop; Aircraft X;' completed the approach and landing without further excitement; and called it a day.Why did I wind up directly _above_ another aircraft on short final? Primarily; I think; because I completely misunderstood the light-sport's arrival path into ZZZ.Reviewing the incident afterwards; it turns out the light-sport plane was not approaching the airport from the north on a left base for Runway XXL; as I thought Tower had told me. The light-sport actually reported their position ten miles south of the airport on its initial call to Tower; and also informed Tower they would be parking in a hanger on the north side of the airport. Tower then then instructed the light-sport to intercept a 3-mile straight-in to Runway XXL.But I don't recall hearing those calls at all. No doubt intelligible sounds came out of my headset and impinged on my eardrums. But I was on short final for my second touch at the time; or in the climb-out on the go; and the picture of an inbound aircraft from the south assigned to my runway did not form on my mental situational map.The light-sport only became an actual plane when Tower told me I was number two for XXL while on the crosswind-to-downwind turn. I am certain that I heard 'on left base to XX left'; but it is entirely possible Tower said RIGHT base and I made a classic 'hear-back' error.Having put the conflicting traffic on the wrong side of the sky; the situation worsened due to several contributing factors:a) I did not inform Tower I did not have the light-sport in sight until it emerged from underneath my nose on short final.b) I did not inform Tower I was unlikely to see the light-sport (or any other airplane in the westerly sky) due to the backlit haze. (Presumably Tower was subject to the same conditions. I doubt they had actual eyes on landing aircraft prior to very short final on either runway; although ZZZ Tower purportedly has a remote display of composited ATC radar returns that works unless targets are too far below pattern altitude.)c) I did not take the opportunity to confirm that the aircraft lifting off from XXL was or was not my traffic. (It turns out turns out it was a Diamond Star--not a light-sport. Its takeoff clearance must have been audible on the frequency; but it evidently didn't register in my brain. I would have known it was _not_ the traffic I was looking for had I listened better.)d) The last link in the near-accident chain was that I considered the absence of a Tower instruction to extend downwind as evidence of traffic no longer being a factor. I do know--I knew prior to this incident--that Tower controllers are _not_ responsible for separation of airborne aircraft. It's all too easy to fall into the trap that the Tower will always keep you out of trouble. I could have filled the hole in the last slice of Swiss cheese by asking Tower to confirm where I was in sequence; never having seen the traffic I was supposedly following. e) It may also be worth noting that at no time did my ADS-B In system announce the close proximity of the light-sport aircraft. My plane's ADS-B In detected aircraft making touch-and-goes on the parallel runway throughout the flight. I don't know why it apparently failed to see the light-sport's ADS-B Out signals. It's a dual-band receiver that doesn't depend on ground-station rebroadcast. For either visual or electronic contacts; one must never forget: absence of proof is ABSOLUTELY NOT proof of absence.Unfortunately; ZZZ Tower and Ground frequencies stream together. When both frequencies transmit simultaneously; whoever spoke first apparently retains recording priority. In this case; Tower's call was apparently blocked by Ground reading a long IFR clearance; then two read-backs were required because the first was stepped on.It's not uncommon for ZZZ Tower to instruct based aircraft with northward parking intentions arriving from the south to be given what amounts to a wide right base; well to the west of the XXR pattern base leg; across the XXR extended centerline then turn right to join final for Runway XXL. My own hangar's on the north side too; I've flown the same arrival more than a few times myself. Had I realized the light-sport was inbound from the south; I would not have looked in vain to the northwest searching for it.