BE-58 instructor pilot reported an NMAC during initial climb. The clearance had been given for the 'option; then runway heading.' The instructor was intending to turn left but forced to turn right. The Tower then questioned the turn to the right. The instructor reports that this tower routinely uses non-standard phraseology and procedures.

Narrative

On Date at approximately XA:45 while approaching the airport ZZZ Tower gave us a clearance of 'Runway XX cleared for the option; fly runway heading on the go'. Upon climb out from Runway XX I saw an aircraft; a single engine high wing model; to the left of me at approximately 200 below. I made a turn to the right to avoid the aircraft. ZZZ Tower then asked us to do a left downwind for Runway XX (opposite of normal traffic pattern operations) and said he wanted us to fly runway heading. Given the traffic conflict which was not made apparent at all from the Tower; I elected to deviate to the right. I then joined a left downwind for Runway XX.I believe that the lack of ADS-B in capabilities in my aircraft inhibited my ability to predict that conflicting traffic. I also believe that the lack of awareness and advisory from ZZZ Tower caused a close call and put minimal separation between myself and the other aircraft. I could have made my intentions known by communicating with ZZZ as to why I was deviating to avoid traffic. In addition; it is my opinion that the Tower consistently uses non-standard phraseology; poor instruction; and often loses sight/awareness of traffic causing an undue increase in these scenarios.