what is considered as cheating?

Problem/Goal: Me F (23) caught my BF M(23) chatting a girl (which is tropa niya) saying things like "nag dinner kana ba?" "kumusta naman araw mo?" and even offered to pick her up from work. (pero hindi natuloy, kasi may sundo na si girl na iba) Is that cheating?

Context: Nangialam ako ng phone ng bf ko which I never do sa buong pagsasama namin (1 year mahigit na kami) cause he made me feel so secured sa relationship namin. Until one day, I did. Totoo nga ano? Curiosity killed the cat LOL. To guys out there, do u usually ask ur tropa na babae if kumain naba sila? Cause when I confront my BF he said na biruan lang daw iyon and ganun lang daw talaga silang magt-tropa. Sa sobrang understanding ko, nagd-doubt ako if ano ba ang cheating sa hindi. I ask him if gusto niya ba si girl, he said hindi. He had this image to me as someone na never magc-cheat, i look up to him, like sobra-sobra. Pero ngayon? Lahat ng galaw niya, pinagdududahan ko na.

Previous attempts: We reconciled. I forgive him but I think I will never forget the pain. He reassure me na he knows his boundaries and will never do things like that again. But everytime na lalabas kami I have this urge na i check phone niya ulit (never ko naman na feel na nagtatago siya ng phone saakin since the day I discovered the conversations) but hindi ko na ginawa ulit, is that the right thing to do?