Ex. "A fateful encounter" or "This Pokemon was apparently friends with a certain trainer." (said in the status part of the PKMN summary)
There are only the ones that you said and some more. "A fateful encounter" "This Pokemon apparently friends with a certain trainer" "Apparently met at (Insert level of Pokemon)" "Apparently arrived at (Insert level of Pokemon)" "Egg recieved" "Recieved at daycare couple" "Egg aparently hatched" "Met at (Insert level of Pokemon)" "Egg obtained" These are the only ways it is said