it sounds like the 'winter games' are calling something the 'white Olympics'

Yes, it sounds like this until you read the next part and see that the case does not match. Then you go back and re-parse. Also, "Olympic games" are inanimate, so you should have suspected something is wrong as soon as you read "называют", which implies animate subject.

why not use себя, or a reflexive verb

These variants can be valid, but they're stronger. They imply the name is the true name.

You can say

olympic games held in winter are called "White Olympics" - a true name for the games.

but you cannot say

"Winter Olympics" are called "White Olympics" - name redefinition.

I do not consider "White Olympics" the "true name" (I've never heard them called that), so I won't use the reflexive verb here.

Now I know that at least some people call "Winter Olympics" the "White Olympics", so I'd write "Зимние Игры иногда называют Белой Олимпиадой"

dative case

I'm not sure I understand this proposal.

it sounds like the 'winter games' are calling something the 'white Olympics'

Yes, it sounds like this until you read the next part and see that the case does not match. Then you go back and re-parse. Also, "Olympic games" are inanimate, so you should have suspected something is wrong as soon as you read "называют", which implies animate subject.

why not use себя, or a reflexive verb

These variants can be valid, but they're stronger. They imply the name is the true name.

You can say

olympic games held in winter are called "White Olympics" - a true name for the games.

but you cannot say

"Winter Olympics" are called "White Olympics" - name redefinition.

I do not consider "White Olympics" the "true name" (I've never heard them called that), so I won't use the reflexive verb here.

Now I know that at least some people call "Winter Olympics" the "White Olympics", so I'd write "Зимние Игры иногда называют Белой Олимпиадой"

dative case

I'm not sure I understand this proposal.

Updates:

In most cases the specific action "называть" needs to be performed by someone [alive]. How can a stone or a game call something something? There are very similar verbs that do not need an animated subject: "Игры вызывают радость." (Games bring the joy.) Bringing the joy is something that a subject can perform even being inanimate.

"inanimate objects can always do things, if only in a make believe story"

That's true. "Труба зовёт в бой" ("Trumpet/bugle calls to battle") is perfectly fine. Each ambiguity has some more common and less common parsing. If you want to use the less common meaning you need to "justify" it by adding details that highlight your logic.

• "Video games take all your time." - I get it.
• "Olympic games steal all attention" - I get it.
• "Olympic Games name the White Games" - sort of grammatically correct, but what exactly does that mean?
• "Olympic Games define White Games as sport games that are played on a white field." or "Olympic Games name White Games the best sport." - grammatically correct, but what exactly do you mean by "Olympic Games define" or "Olympic Games name"?

The incompatibility between inanimate subject and "называют" does not extend to the reflexive form "называются". It seems to me that the "Inanimate Subject + Reflexive Verb" expressions are most often understood as passive. The specific reflexive form "называются" is now biased towards passive connotations even when the subject is animate. "Называют себя" ~= "call themselves", but "называются" ~= "are called" (even though these were the same words in the beginning.).

Parsing:

1. "Игры" => Inanimate (Nominative Subject OR Accusative Object);
2. "называют" => commonly needs animate subject => "Inanimate_Subject" parsing of "игры" is unlikely, the likely parsing is "Inanimate_Object in an indefinite-personal sentence"
3. "белой олимпиадой" => not Genitive case (not "Subject calls object .") => Likely parsing is "Object is called name", but still not 100% sure - it could still be "Subject defines as description."
4. End of sentence => "Subject defines name as description." parsing is broken. This leaves the "Object is called name" variant.