Leptospirosis is a bacterial infection spread through contact with animal urine or water and soil contaminated by animal urine.[1]
Rain falling from the sky isn't contaminated by animal urine, so it's safe in that regard.
On the other hand, the rainwater could cause existing contamination in pools and soil to spread over a larger area, increasing risk of contact.
In urban areas rain is typically drained away quickly into stormdrains or the sewer*, so I expect it's likely to leave things cleaner rather than more contaminated.
However, if it rains so much the sewers overrun, then the risk of leptospirosis adds another reason not to play in the water. [1][2]
But I suspect none of us really needn't an extra reason not to play in sewer water.
*) While not recommended by the WHO[3], combined systems that handle both rainwater and sewage still exist in many places.
To address the updated question about rain puddles, we probably need to get a bit speculative, since I can't find research directly addressing it.
One part of the equation is the likelihood that a rain puddle would be contaminated. In a city this will mostly depend on the presence of rats, which are the most common cause of transmission[4]. Places where they live or eat are at high risk of contamination (presumably the places in between less so). So puddles in a park might be at risk, because parks are a good place for rats to make burrows. And places with a lot of litter/garbage are a risk, because that's where they can feed.
The other part of the equation is the likelihood of infection from interacting with a contaminated puddle. The bacteria can enter your body through open wounds or mucous membranes (eyes, nose, mouth)[4]. I don't think this is likely to happen when running through puddles, except perhaps if a car splashes you top to bottom. I suspect the biggest risk in this scenario is after getting home and taking off your wet running shoes, forgetting to wash your hands, and then eating something.
To give some sense of the total risk: in New York City there were 44 locally acquired cases between 2006 and 2020, and 13 in 2021 up to September [4] (and an increasing trend). So on a population of over 8 million the risk is fairly small, but not zero. Using that last number, it's roughly as likely as getting struck by lightning [5].
[1] "Leptospirosis from water sources" PMID: 25348115
[2] CDC: "Hurricanes, Floods and Leptospirosis"
[3] WHO on storm water drainage
[4] NYC Health, "2021 Advisory #35:
Increase in Leptospirosis Cases in New York City"
[5] CDC, "Lightning: Victim Data"
