Piper PA-18 pilot reported an NMAC event shortly after takeoff due in part to inaccurate position announcements by student pilot. Piper Pilot contacted the flight school to learn from the incident.

Narrative

I called FSS on the CTAF frequency of XXX.X for an airport advisory as I was taxiing on Taxiway XX. The FSS responded there were two aircraft in the pattern. I continued to monitor the CTAF of XXX.X while taxiing on [taxiway 1; 2; and 3 to Runway XX/3 intersection where I held short of [Runway] XX. I performed my engine run up; and not hearing anyone on base or final I announced I was taxiing onto Runway XX at 3. I heard a Cessna announce they were holding short of XX; and before I entered the runway I visually looked to the approach end (North) of XX and saw the Cessna holding short of the runway. I did not see any other traffic and taxied onto Runway XX at taxiway 3 (intersection takeoff approximately mid field). When I was on the runway I announced I was departing Runway XX at taxiway 3. I heard a lady in a Cessna announce she was 'Upwind'. My plane was lightly loaded which resulted in a short ground roll and good rate of climb. I was quickly off the runway and near 100 ft. AGL when I looked to my left and noticed a golfer looking at me (the airport is adjacent to a golf course). I then saw a blue and white Cessna in very close proximity to me on my left side about 100 ft. to the left and 50 ft. lower. I called on the radio and asked if she had announced being on final. She replied that she had also announced 'upwind'. I replied that it was a close call and we both needed to be more vigilant. I landed at a nearby airport and called FSS and asked for him to pull the tapes. He contacted ZZZ FSS where the tapes were recorded and called me back. He reported that the Cessna had reported turning final about the same time I called up for an airport advisory on the same frequency. He also reported that she announced her position as 'upwind' about the time I took off. I reviewed the incident with a very experienced flight instructor as well as reviewing the AIM for proper traffic pattern positions and intersection takeoffs. We determined that the lady flying the Cessna had improperly reported her position as 'upwind' when she was over Runway XX and was on the departure leg. Also contributing was the large pattern flown by the Cessna. There was enough time from the time she announced turning final for me to receive a briefing; taxi to the runway intersection with taxiway 3; perform and engine run up; taxi onto the runway; and takeoff. The instructor suggested I perform a more thorough look up the runway turning 45 degrees to get a better view prior to taxiing onto the runway. He also commented that it sounded like I did everything right and that there was nothing wrong with the intersection takeoff that I performed. While I did look to the approach end of the runway prior to taxiing onto the runway and saw the other Cessna holding short; I did not see the Cessna in question which must have been nearing touchdown or on the runway.I have contacted the flight school who owns the Cessna hoping to discuss the incident so we can all learn from it and improve safety but have not talked with them yet.This was the first flight of the day for me and I was well rested.