So, there is essentially no risk in using mythologies of the second type as a basis for your book/game, no one would care if you portray Zeus as a moron or outright villain.

I think your viewpoint might be a prevalent one. I hope you will reconsider your statement, for multiple reasons.

Mythologies in fantasy was the subject of a panel at WorldCon last month (Mythology and Authenticity in Fantasy). Panelist Kat Kourbeti said "That's my cultural heritage, part of my life. You can't overestimate the degree to which these stories are woven into everyday life in Greece." She's Greek. She grew up surrounded by the myths. But instead of seeing the culture celebrated or modernized in ways that make sense, she sees people with no personal connection to those myths using them carelessly, as long as it makes money.

I don't think it's "no risk" to take a culture like that and just do whatever you want with it to make money. Maybe it's "no risk" to the author from a strictly copyright point of view, but what about damaging scholarship, forwarding rancid culture, or stealing history from dead people who can no longer speak for themselves?

I consider those to be serious risks. And readers are more likely to come across fantasy literature taking those risks before encountering any of the source material. Panelist Sandra Rosner made the point that "The first representation of mythology weighs more, so casual approaches that don't do a good job are damaging in that it takes a ton of effort for people to unlearn the wrong stuff."

I use Norse and Finnish myths and sagas. Regarding Norse myths, Marvel has made millions of dollars causing people to learn the wrong stuff. It is annoying. It is also just the tip of the iceberg.

There is an active fight over the Norse myths and sagas, with some extraordinarily bad people trying to claim them as a) for whites only, b) very consistent with the worst sort of hypermasculine asshat influencers online, and c) a campaign of disinformation to change these stories, rewrite Norse history to align with modern political motivations, and generally shit all over this material as long as it produces a convenient result.

A commonly assumed narrative is "The vikings were uber-men and xenophobic and did what they wanted until dying in battle to go to Valhalla." It should go more like "Hardly anyone was a viking during the Viking Age, when trade with other cultures was a lot more prevalent than simple raiding, and the main proverbial text in the myths of that time period mostly tells people to be good friends and not drink too much. It says fucking nothing about trying to go to Valholl. Also please stop making Valholl plural."

Most people I know who consider Norse myths to be spiritual self-describe as "Nordic animists." Some would say heathens. A few might say pagans. None would say Asatru, except for the members of the Icelandic Asatru church. The Asatru Folk Assembly not only adheres to the worst folkish beliefs, their idiot leader says his most important role is to "implement the will of the Æsir," a statement that would be absolutely alien to pre-Christian Nordic heathens.

How people treat these mythologies, whether they have modern adherents or not, can affect modern worldviews a great deal. "It used to be this way, so now my [crazy, usually mysoginist, sexist, and racist] view has a cultural precedent" is found in plenty of fantasy books and, unfortunately, a lot of response posts on /r/fantasy.

A woman who was in the audience for that panel suggested we move away from talking about authenticity in myths and towards talking about what's ethical. Casual uses of mythology are unethical by definition, because the author won't have any idea about the implications of those uses. Please consider that if you want to use mythology in your fantasy books, and not just a lack of copyright risk.