What is persuasive speech?
It is when you want to make others believe that what you are saying is totally true. For example, let's suppose that you are delivering a speech about "getting 10 years younger just by taking a pill for two weeks". Your audience may think that that is utterly impossible; nevertheless, you have to make them change their mind by using all your strategies, so that your audience, or part of your audience, will believe that you are right.
Both in a debate and a persuasive speech you have your side and you are presenting it in the most favorable light. Regardless of the format, you are essentially persuading in both circumstances. Your efforts are solely devoted to trying to have a person, judge, teacher, etc. see the merit in your side, so in that, when in a debate, you are persuading, and your speech is essentially the same as a persuasive speech.