Didn’t read the entire thread, but there are terms in use already. EU (or European) Delay, as opposed to US Delay.
The two styles are not functionally equivalent although they deliver the same amount of time move-b-move per Tom’s point. (Bottom line, that does, in a sense mean they are functionally equivalent, but I think we all understand that important details differ between the two methods.)
However, the functional differences do have an impact in terms of display.
Bronstein/EU Delay displays on a single display the total amount of time remaining for the player’s current move or if it is the opponent’s move, the player’s very next move.
US delay displays on a dual display, one for delay one for the main time, the total amount of time remaining for the player’s current move, or if it is the opponent’s move, the player’s very next move. The dual display for US delay may not be visible on the player’s display except when the timer is counting down the delay time.
Both of these essentially say what is there currently. But allow for a slightly different construct.
In delay, a player receives an added amount of time per move of time used to a maximum of the stated delay time.
In increment, a player receives an added amount of time per move of the stated increment time.
Increment is a set addition per move, delay is the lesser of time used or a stated addition.
Defining it this way allows the display explanation without further description of what gets added back or used etc. So it should cut it down:
[size=115]Delay and Increment:[/size]
In delay, a player receives an added amount of time per move of time used to a maximum of the stated delay time.
In increment, a player receives an added amount of time per move of the stated increment time.
Display
Bronstein/EU Delay displays on a single display the total amount of time remaining for the player’s current move or if it is the opponent’s move, the player’s very next move.
US delay displays on a dual display, one for delay one for the main time, the total amount of time remaining for the player’s current move, or if it is the opponent’s move, the player’s very next move. The dual display for US delay may not be visible on the player’s display except when the timer is counting down the delay time.
Increment is always displayed using the Bronstein approach.
(I can’t think of an example where the last statement isn’t true - is anyone aware of one?)