Couple questions. You say that it's not the BU's job to give the count. Do you also mean in between pitches when a fielder or runner turns to you and asks for the count, as the BU you should ignore it or refer them to the PU?
On the topic of keeping the count, I was on another site and a guy (don't know his credentials or what levels he was referring to)was saying that when he worked the plate he never bothered using a counter. Others chimed in and some used them and some didn't, it was strictly a personal preference. But this gentleman went on to say that if his partner working the bases was using a counter, he would tell him to stick it in his pocket and he didn't want to see it again. He claimed that it was insulting as the PU for the BU to be using a counter. Is that common practice at levels above high school? Also, is tapping the top of your head a proper mechanic for asking your partner to confirm the count?