I am a bit confused about differences in the usage of articles before ordinal numbers. Technically, it would be logical to always use "the" for instances like "the fourth time I did this" or "the tenth city to host this conference", but I've also encountered such phrases as "The show was renewed for a second series." How does this work?
The use of the indefinite article for a planned or commissioned second series of show A works for two reasons.
Firstly the planned series is one of a number of planned second series. These planned second series are not all of show A but second series of show B, show C and so on can all be at the planning and commissioning stage at the same time.
Secondly plans for a second series could be dropped for financial or other reasons before it is broadcast and, perhaps, the plans revived in a different form later. In that case the newly commissioned second series would become the second series and the original second series would never be broadcast.
We use the same structure in other contexts as well, people say things like "I'd like a second helping of dessert"; "I'm looking for a second car" and even "We're trying for a second baby". Before the dessert is served, the car is bought, or the baby is born these things are intentions or aspirations and become concrete only later.
When we look back at these things afterwards we use the definite article and say things like "The second helping of dessert was even better that the first, it had more fruit in it"; "The second car is a two-seater sports, it's more fun than the people carrier but less practical" and "Unfortunately she lost the second baby at 25 weeks, it was another boy".
If, in the last case, the lady has a third, successful, pregnancy resulting in a daughter we say "Her first child is a boy and the second is a girl". The girl is the second child even though there was a stillborn baby between her and her brother. The parents will remember, and may even continue to mourn, the stillborn child but he is not usually counted in the sequence of growing and later adult children.