A General Aviation pilot reported confusion regarding an IFR cancellation that was not acknowledged by the Controller prior to switching to CTAF. This miscommunication resulted in the Center not receiving the pilot's IFR cancellation.

Narrative

I was on an IFR flight plan from ZZZ1 to ZZZ. Weather at ZZZ was clear and 10 but I elected to do the RNAV XX approach to continue to build proficiency with my GFC500 autopilot. At the FAF (ZZZZZ) the ZZZ Center Controller advised there was one plane observed in the pattern at ZZZ; to cancel IFR in the air or on the ground at ZZZ; and change to the CTAF. I observed the reported aircraft via TIS-B; and transmitted to the ZZZ Controller that I was cancelling IFR and would 'squawk 1200.' I thought I heard the ZZZ Controller acknowledge my IFR cancellation (my 'Expectation Bias'); and I changed to the CTAF frequency; communicated with the aircraft in the pattern at ZZZ; and continued on the approach. Subsequently another aircraft came on the CTAF frequency and asked if Aircraft X was on the frequency and stated ZZZ Center had not received my IFR cancellation and did I want them to relay my cancellation. I was surprised by this request because I thought I'd heard the Controller's acknowledgement (again; my 'Expectation Bias'). I replied in the affirmative to the third aircraft and a minute later they advised they had successfully relayed my IFR cancellation [to] ZZZ Center. The two lessons learned are: My focus on a potential traffic conflict allowed an 'expectation bias' of the Controller receiving my IFR cancellation. Second; don't use the phraseology of 'Aircraft X is cancelling IFR and squawking 1200.' Instead; only use the phraseology that 'Aircraft X is cancelling IFR' and wait for the Controller's acknowledgement and instruction to 'Squawk 1200' to be sure the Controller did in fact receive the IFR cancellation.