I played around with the code and I finally found a way to do what I wanted. In the meanwhile, I think I've also better understood the behaviour of Manipulate.
This is my solution; it deals with the second MWE I posted:
Manipulate[
(* === Calculations that are triggered each time there is a change
in tracked symbols —'parameters', in this case: === *)
message = "Updating..."; result = "";
Pause[2];
result = parameters[[1]] + parameters[[2]];
message = "Ready";
(* === Results to be shown: === *)
Column[{Dynamic[message], Dynamic[result]}],
(* === Controls: === *)
{{a, 5}, 0, 10, 1, Appearance -> {"Labeled"}, AppearanceElements -> None},
{{b, 5}, 0, 10, 1, Appearance -> {"Labeled"}, AppearanceElements -> None},
Button["Calculate", parameters = {a, b}],
(* === Other settings: === *)
TrackedSymbols :> {parameters},
Initialization -> (
Clear[a, b, parameters, message, result];
parameters = {5, 5}; result = 10; message = "Ready"
)]
This is the result:
