Pilot reported an NMAC event during landing pattern practice with another aircraft entering the pattern for different runway.

Narrative

While completing takeoffs and landing within the traffic pattern at ZZZ I was involved in a near midair collision with Aircraft Y [Piper]. The Piper was entering the pattern at the same time I was preparing to turn downwind from crosswind. As I began turning a left downwind for Runway XX; the Piper was in front of me and above about 100 feet. I observed the Piper through my windshield and my TCAS system alerted that the traffic was within 100 ft. I made a radio call; leveled my altitude; and made a radio call to alert the Piper and continued to landing as normal Runway XX. The Piper followed in behind me without incident. I believe the Piper meant to enter a 4 midfield left downwind; but entered the pattern much earlier than intended. The Piper was also above me in a low wing airplane and descending into the pattern; and I don't believe the Piper ever saw me until I made a radio call. In addition; the radio call outs from the Piper seemed non standard and vague; and one even mentioned 'ZZZ1'; making it difficult to pinpoint the Pipers location and intentions. What could prevent this is a standard pattern entry from the Piper; and being at pattern entry altitude well before the pattern entry would have made it more likely the Piper would have seen me. Standard radio callouts are very helpful. I will begin making radio callouts during the crosswind leg. I will begin leveling out and flying a square crosswind to downwind turn to increase my own visibility to entering traffic.