Too many choices

swim    have a swim    go for a swim    go swimming     go to swim

In recent post, I discussed V and have a N, specifically argue and have an argument. In a comment, I added go for a swim. Later, I also thought of go swimming and go to swim.

In general, the first four seem to be interchangeable, but the last may have a different nuance.

I swam    I had a swim    I went for a swim    I went swimming    I went to swim 

The first four entail that I did actually swim. The last doesn’t (automatically): I went (somewhere) with the intention of swimming. In fact, the third might also mean that I didn’t swim. I can think of a difference between We went-for-a-swim and We went (to the beach) for a swim(, but it was closed because of coronavirus restrictions). 

Consider also:

I swam at the Olympics    I had a swim at the Olympics    I went for a swim at the Olympics    I went swimming at the Olympics     I went to swim at the Olympics 

The first definitely means that I was a competitor. The fifth might mean that. The others probably mean that I was a casual swimmer. This difference probably has more to do with the requirements of swim at the Olympics, compare I sang at the Olympics. (True: I was in the massed choir for the opening ceremony.)

I haven’t been able to find go for a swim in the Cambridge Grammar of the English Language. Have a swim is a light verb construction, and go swimming and go to swim are catenative verb constructions, but what is go for a swim? It has some similarities with both, and is obviously a unit of meaning by itself; compare I went for a pizza. The indexes don’t help. I looked under go, for and swim, and the grammatical index doesn’t really help unless you already know what a construction is called (and GCEL often calls things by different names than everyone else). Maybe I’ll let serendipity guide me to the correct entry.