This really comes down to a matter of your business model. Frederick, I really don't know your industry well enough to know how most business coaches handle the payment structure. What I can tell you is the half down, half when complete (or divided into thirds) model is standard in our industry. There's no rule that says we need to do it this way, but it is a common practice. It's so common, I call it standard.
There's also something about our industry that makes us highly susceptible to getting stiffed if we don't use this model. To explain why would be little more than speculation, but every experienced designer has found this out. The down payment really does work because anyone who has a problem with it usually has a problem paying at all. It is rare for anyone to question the down payment, or at least I've not had that in a long time. It could be that people who don't like to pay easily find those cheap providers all over the Internet so the rest of us don't see them much.
As for the trust issue, that's not a problem either. I'm rarely approached out of the blue. It either comes as a result of my online presence, or, if they do just happen to find me, they do some research, then contact me and often tell me they feel comfortable because they researched in advance. Either way, the trust is built ahead of time.
With that said, I began testing a different model a year or so ago, and it's been working well. The fact that the 50/50 model is standard got me thinking that being different could be interesting. I'm currently making some other changes to my company so I can better implement the new model. There's a significant potential to get too much work too fast, so I need to plug that before it springs a leak. That's a good problem to have, but not if things spiral out of control.
But I fully get what vangogh is saying. It's not so much about distrusting everyone, there's just something about our industry that invites being taken advantage of if you don't put the proper checks and balances in place. Experienced designers know that, and that's why almost all the best design people require a third to half down.