Once, I began speaking with Saotome Shihan. I didn't bow because I had a cigarette in my hand. I hope to this day that he did not take it as disrespect since that's exactly what I wanted to avoid. I said, "Hello, Sensei," partly because it's more formal, and partly because I didn't want him to mistake "Hi" for "Hai." What's the general Japanese rule on this? Is it okay to bow with cigarette or anything else in hand? Fortunately, he didn't get into a lecture on the subject, but these East-West differences can be confusing to me. I suppose I just went with my heart by not bowing.
FYI, Saotome Sensei knows you're not Japanese and doesn't expect you to try to be something you are not. He's lived in the USA for 30 years, has been a citizen since, I believe, the 1980s. He very often responds to bows with an outstretched hand, and I don't believe he is trying to say "you can't bow right" or anything like that.