Captain reported lack of clear communications by Aircraft X pilot resulted in taking evasive action on climb out; and a NMAC event.

Narrative

Conditions included winds 160 at 8 knots; ceiling broken about 2;200 feet; visibility 10 miles. I had just started the engines when I saw two Aircraft X banner tow aircraft taxiing to Runway XX for departure. Minutes before; Aircraft Y had landed on Runway XY; after what I assume was a localizer approach. I checked the automated weather; and then followed the Aircraft X to Runway XX. During the taxi; I reported that I was taxiing to Runway XX behind the Aircraft X. The Aircraft X pilots were communicating with their ground crew on the CTAF; and they indicated they would be 'looping around' to Runway XZ for a banner pickup. In my recent experience; the banner pickups had been conducted on the western portion of the airport; in the grass on the south side of Runway XZ. Also in recent days; when appropriate; the banner tow pilots had typically been instructed to operate as to not interfere with arrivals and departures on the cross Runway. As I completed my run-up and pre-takeoff checks; the Aircraft X departed. I reported taxiing into position and departing Runway XX behind the Aircraft X. I estimate I departed about 15 seconds behind the second Aircraft X. I knew the Aircraft X would be maneuvering to Runway XZ; but did not expect them to make a low pass over Runway XX until all departures had cleared the area. It was unclear how they were intending to 'loop to XZ;' as they normally stayed clear Runway XX during these pick-ups. To my surprise; the Aircraft X were maneuvering by making a low-level left turn to line up with Runway XZ; overflying Runway XX in the process; and apparently were not aware of my departure. Shortly after rotation; while I still had the second Aircraft X in sight ahead of me; I glimpsed the first Aircraft X northeast of my position; at low altitude; north of Runway XZ; apparently maneuvering for his pickup. At this time; the Aircraft X pilot must have seen me; and altered his flight path. A ground crew person in charge of the banner tow operation became agitated and yelled over the CTAF asking who had departed and caused his banner tow airplane to alter course (not in those exact words). I responded with my identification; and said that I departed behind the two Aircraft X. Later that afternoon; I spoke with the head of the banner tow operation; who insisted that I should not have departed; but instead should have waited until the banner tow planes had completed their maneuvering and banner pick-ups. I suggested that I was following standard procedures; had reported my intentions to depart behind his two aircraft; and had followed the second Aircraft X at a safe distance. I also pointed out that it was unclear exactly how those pilots would be maneuvering for their banner pick-ups; and that I was not expecting a low-lever overflight of Runway XX when that was the Runway in use. I am concerned that the low-level maneuvering of the banner tow operations in a non-standard manner creates an unsafe situation when the banner pick-ups use the cross Runway; and overfly the Runway in use at low altitude. This airport is non-towered; and sees many transient aircraft and less experienced pilots (including students) that would not be aware of the unusual procedures followed by the banner tow pilots. The use of non-standard phraseology (i.e.; 'looping to Runway XZ'); rather than more explicit and standard phraseology; contributed to the confusion. I would suggest that when winds are not excessive; pilots could be using the same Runway for takeoff and arrival as the banner pilots are using for their banner pics. With all aircraft using a similar pattern; and not maneuvering for a crossing Runway; the potential for low-level conflicts would be reduced. I know from years of instructing that when using a cross Runway (e.g.; for crosswind training); the pilot should remain situationally aware of the other aircraft maneuvering for the main Runway in use; and take steps to avoid conflicts with aircraft performing normal operations. This is a common courtesy and safety practice.