The Church of Scotland is presbyterian. 'Presbyterian' and 'Church of Scotland' are not mutually exclusive. Presbyterianism is not a set of beliefs, but a system of church organisation, which involves tiers of committees, as opposed to tiers of bishops as in the RC and episcopalian churches. There are and were umpteen different presbyterian denominations in Scotland.
Three kirks have been mentioned in this thread.
St Kentigern's was the ancient parish kirk of Lanark. It was abandoned in the 18th century and in ruins by 1834. See
https://canmore.org.uk/site/46576/lanark-hyndford-road-st-kentigerns-churchSt Nicholas' was built in 1774 to replace St Kentigern's, and is still in use. See
https://canmore.org.uk/site/46586/lanark-high-street-st-nicholas-churchThe account of Lanark in the New Statistical Account states specifically, 'There is no chapel of ease attached to the Established Church, although one is much needed, especially at New Lanark'. See
http://stataccscot.edina.ac.uk/static/statacc/dist/viewer/nsa-vol6-Parish_record_for_Lanark_in_the_county_of_Lanark_in_volume_6_of_account_2/ (page 26).
The 'Established Church' in this context is the Church of Scotland.
Chapels of ease were built when the population of a parish became too large for the capacity of the parish kirk, or when a new town or village grew up which was distant from the parish kirk. The statement that there was no chapel of ease attached to the Church of Scotland parish kirk means that there was only the one Church of Scotland in the parish of Lanark in 1845, and by implication in 1833. Therefore you can be confident that your baptism record was written down by the clerk of St Nicholas' Church.
The item on Greyfriars Church posted by
goldie61 makes it clear that the history of churches in Lanark is not quite so simple, with a whole raft of churches set up and later closed or amalgamated with one another, almost all of them later than 1833. (It also contradicts the NSA by saying, 'Looking back slightly to 1840, St. Leonards Church was opened as an extension church to the Parish Church', but that is still after your 1833 baptism. I'm inclined to believe the minister who was writing in 1845, rather than someone writing 150 years later, but fortunately it doesn't affect your question!)
However, it is important to understand that
it was the exception rather than the rule for weddings to be held in the kirk building. The traditional place for a wedding ceremony was in the bride's home. If she had no parents, or was marrying a long way from home, the ceremony might be held in her employer's house, or in the manse. Only towards the end of the 20th century did it become popular to marry in a hall, restaurant or hotel, and kirk weddings did not become the norm until the beginning of the 20th century.
Likewise baptisms were often conducted in the parents' or grandparents' home. Unless a record states specifically that a child was baptised before the congregation, or that the baptism was witnessed by the congregation, you cannot assume that the ceremony took place in the kirk. (I myself was baptised in the best room of my parents' house, so home baptism still occurred well into the 20th century.)
Also you have to bear in mind that in theory the Church of Scotland was supposed to record all baptisms and marriages, not only those of its own adherents, and sometimes it even actually managed to do so.
To summarise, (a) SP doesn't need to state which kirk the OPR refers to, because by definition it's the parish kirk and (b) you cannot assume that a wedding or baptism recorded in an OPR was conducted in the parish kirk, or indeed in any kirk.
