"Processing" is both a "state" (major process step) and a "status" (minor information label within a step). An order goes through several states and each state has several statuses, a default one and other options. When you are in a particular state, in that comment/modification box, you only will be shown other statuses that are "linked" to that state.
It is an annoyance indeed that many credit card type payment modules go straight to "processing" state but the default magento ones only allow a choice of "new" state. This cannot be changed 100% except programatically but you can get close without coding.
First option (module): use a "bank prepayment extension" which allows to pick a state/status from the processing state. There is a free one I use does, but it also allows all the way to closed state so I don't trust it and go to a new state with it. But I didn't test it). If you pick this, i still recommend tweaking the statuses as in step 2
Second solution (configure statuses): You can add new statuses, and you can "assign" these statuses to one or states, to reflect your logic.
In my case I have people who pay "up front" with paypal, cards, as well as people who pick "invoice me" or "bank payment". I was struggling because once you invoice orders, it's hard to tell what has and hasn't been paid. As I only preauthorise payments, it's also hard to track if something has been "charged" or not.
I made a different choice as you, in that I chose to "invoice" but have a different status available before and after "invoice" to track not yet paid. Just create your own flow.
So I added more statuses to the "New" and "Processing" stage. There's an "awaiting payment" state and status which is only used by certain payment methods, so I decided to add this status to my New method. There's also a "still needs to pay" option in "Processing" to really make it obvious. Also, I have the "preauthorized" status as the default one for card payments, as reminder that they need to "invoice" to trigger the charge. Also added some statuses to mark orders on hold and problems...
I'm still tweaking it over the months, and once it's just right I might explore coding to make the "flow" more rigid but that's something I am wary to do.
Just to illustrate, here's my list.
STATE [New]
New Pending pending
Confirmed confirmed
Awaiting Payment pending_payment
STATE [Processing]
Pre-authorized preauthorized
Processing processing
Still Needs to Pay not_paid
STATE [Pending Payment]
Awaiting Payment pending_payment
STATE [On Hold]
Preorder Hold preorderhold
Problem problem
On Hold holded
STATE [Complete]
Follow Up followup
Problem problem
Complete complete