That can be true or false. If someone is intending to reference tears of sadness, then that would be a short, appropriate way to say it though it is uncommon. When someone is intending to reference tears of laughter, they will more commonly say, "I laughed so hard I cried." It can also be used to reference tears of joy (think a wedding proposal). In that case the statement you used could again be correct, but you might also hear them say, "These are happy tears," or "I was so happy I cried."

"I'm in tears" can also mean, "I'm crying" with proper context, you can also say, "I'm weeping, I'm sobbing, I'm shedding tears."