A budget in which the resources are variable except that a deadline or duration is fixed.

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The term "budget" is usually constrained to meaning "what you spend", excluding the pure notion of time, which is usually called "schedule".  "Fixed Time" budget doesn't describe a budget any better than "Variable" budget.

A project can be constrained to a fixed budget.  It can be constrained to a fixed schedule.  It can be constrained to delivering a fixed amount of value (the hardest of the three to measure by far).  It can't be constrained in all three dimensions at the same time, unless the constraints are artificial.

Fixing the time of a project leaves the other two dimensions to trade off against each other.  In a month, you deliver more value if you spend more, but the relation is not linear, so if you're already spending "enough", additional spending won't help.  Many software managers have a hard time understanding this.

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I think you may be confusing "budget" with "actual."  I cannot imagine anyone starting any project without a plan to produce a certain set of functionality within a certain time frame within a specified cost.  The difficult part is deriving your budget from your estimates or adjusting your planned deliverables to fit your budget and schedule, but this is required to start.  No one is going to issue a blank check for an unspecified deliverable at an unspecified time. 

''I cannot imagine anyone starting any project without a plan to produce a certain set of functionality within a certain time frame within a specified cost.''

Happens all the time.  Development proceeds from sketchy requirements with no plan at all.

So where did the development team come from?  Someone decided to assign some number of people with certain skill sets to accomplish something.  "Sketchy" requirements are vastly different than no requirements.  The constraints of budget, schedule, or intended functionality do not need to be restrictive to exist.

''No one is going to issue a blank check for an unspecified deliverable at an unspecified time. ''

Sometimes adjusting the budget (upward) is not an option, in which case the budget is the actual.

The actual cost will never equal the budgeted cost.  It will either be higher or lower.  Depending on the amount of risk (of exceeding the budget) you are willing to accept guides how you set the budget.  In a firm, fixed price contract, you will set high budgets to minimize the risk of overrun, but you do not expect to meet the budget if everything goes according to plan.

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I run into FixedTimeBudget often.  The form I see is the sales force has promised a certain feature set by a certain date, and management does not have the backbone to correct this in front of the customer. --PeteHardie