Trust is when you let someone do something that only you feel comfortable personally doing or is best done by you.. and that you want to do. If I win the lottery with a ticket, I'll be trust if I for some reason let someone go cash the ticket on my behalf. It's better for me to cash in the ticket because they could steal the money after.
Trust really ultimately defined is when you let someone do something you're not comfortable with them doing, and giving them access to do so. Letting your girlfriend go visit her ex-bf for example assuming she's cheated once before.
Lots of people have all sorts of things on their laptops or computers they don't want anyone to have access to, as a result, they wouldn't share their computer password with anyone. If I gave someone my password to access a certain file and nothing else... that's trust because I'm letting them do something that only I feel comfortable doing... and that's accessing my computer, or rather, I'm letting someone do something I'm generally not comfortable with them doing.
This Black Maria situation would only be trust if Sanji was already fighting her and thought he was the only one capable of defeating her in this situation, and Robin requested that Sanji stop fighting and let her takeover. If Sanji didn't want to, but let her do so? That's trust.
If that's not the context, it's not trust. If I have to go knock on someone's door and I'm scared to do so, or don't want to do it... me passing on this responsibility to another person isn't trust, that's just being a coward.
Just because someone says thank you for trusting me doesn't mean you trust them especially when they lack context. If I don't want to go knock on someone's door because I know there's a dog in there, and I let someone do it on my behalf and they say "thank you for trusting me", it's not trust, it's just me being a coward or passing on my own problems to another.
He simply shifted his problems to the last person that should be involved in that situation which makes it worse.
There's a place for trust, and this situation wasn't it.
Sanji didn't want to fight Black Maria, and that's all that's needed to understand that this had nothing to do with trust.