To be honest, I do not think that you should have to make your T understand that you can be trusted and that you need help with the things you said. I rather think that should be the obvious thing a T does, unless they know you very well and can judge that you are just trying to get around talking about the real issues or something like that. And since this is a new T plus you said that wasn't the case, they should let you take the lead on what needs to be addressed.

As you said, this T was your best option so far. I don't know what your options are, personally I'd try searching for someone else. But if you want this T another shot, maybe go back and clearly state what bothered you. Tell them that you find them to be the best option so far, but that they seemed very dismissive and not understanding of your situation. Tell them clearly what you'd like from them and why. Maybe also ask for the reasons why they think something else should be worked on first, maybe there is a good reason for that suggestion.

Finally, about your therapist acting cool when you tell them you think about suicide, I think that might be just the Ts way of wanting to show that they are not repulsed or scared by this statement, since that is oftentimes the reaction that people get, which is not a very helpful reaction. But I can of course see why it would be bothersome.