The more I research this, the more confusing it gets. One problem is that the terminology is apparently not the same in all countries, and different sources give different definitions.

The first thing that seems to be clear is that "condicional" and "hándicap" are not the same thing, as Anthony's reference already indicates. The second is obvious from the source texts you've quoted: "terreno condicional" denotes a lesser type of race for a horse that has not yet been very successful. When it starts to be successful it competes "fuera del terreno condicional". If it doesn't do well, it may return to the "terreno condicional", like Cirque du Soleil in the second example you've quoted.

First, here are some official definitions from the Argentine Reglamento del Turf:

"V – Carrera de peso por edad: es aquella en que los caballos llevan un peso proporcional determinado, establecido en la escala correspondiente de acuerdo con la edad de cada uno. Conserva esta misma designación cuando las condiciones estipulan recargos o descargos.
VI – Carrera condicional: es aquella que no es a peso por edad ni handicap, y se establece una condición a cumplir por los inscriptos.
VII – Handicap: es la carrera en la cual los caballos llevan un peso fijado por el handicapper con el fin de igualar las posibilidades de ganar de cada uno."
http://www.purahipica.com/export/sites/purahipica/descargas/...

Now, there is a type of race called a Conditions Race:

"Conditions races are horse races where the weights carried by the runners are laid down by the conditions attached to the race. Weights are allocated according to; the sex of the runners, with female runners carrying less weight than males; the age of the runners, with younger horses receiving weight from older runners to allow for relative maturity, referred to as weight for age; and quality of runners, with horses that have won certain values of races giving weight to less successful entrants.
Conditions races are distinct from handicap races, where the weights carried are laid down by an official handicapper to equalise the difference in ability between the runners."

But the trouble is that conditions races, at least in UK terminology. are the most prestigious races of all: the Epsom Derby is a conditions race, for example:

"Conditions races are staged at all levels of horse racing. As all of the most important races in Europe are conditions races, the term may also refer to the races for the very best horses, known as Group races."
http://en.wikipedia.org/wiki/Conditions_races

This clearly doesn't fit the sense of "condicional".

In Chile at least, "condicional" means a race in which the horses have to be of a certain type: to fulfil a certain condition. They are restricted. There may also be a weight allowance according to the type of horse. This fits the Argentine reference, which says "se establece una condición a cumplir por los inscriptos":

"Existe en Chile dos criterios definidos para clasificar las carreras, además de los clásicos: Condicionales y Handicap.
El primero se refiere a que todo caballo para estar inscrito en la carrera, debe cumplir con una determinada condición dada por la edad, sexo y carreras ganadas; por ejemplo ser hembra de tres años no ganadora o macho de dos años ganador de una carrera. El segundo se refiere a una clasificación por campaña, donde los caballos obtienen un determinado índice." (p. 149)
http://books.google.es/books?id=XP9GRkGbrJYC&pg=PA61&lpg=PA6...

I think the best equivalent is "allowance races":

"Allowance Races
Allowance races are exactly like their name implies. Allowances are made or "conditions are set" in order for the horse to be eligible in that race. Examples of allowance races are: Non-Winners of 2 (races), Non-Winners of 3 (races), Non-Winners of 4 (races). As you can see, each level is more competitive. A horse that has never won two races might have a hard time winning a race against horses that have won three."
http://www.sportsbettingacumen.com/horse-racing-articles-typ...

"Allowance Race – This is a non-claiming race intended for younger horses not yet ready for stakes competition or horses that are not for sale. Such races may be limited to non-winners or limited to entries that have not yet won one, two, or three stakes races. Weight allowances, such as three pounds, may be applied based upon recent track records or age-sex differences."
http://www.betbind.com/help/articles/all-types-of-horse-race...