How do I use 2 different taxonomy view modes on one node page?
The problem is that the view mode is also part of the cache keys and you would need to update them as well to avoid that both view modes are cached under the same cache id.


  Is there possibility to disable render cache for specific node and/or
  fields?


Yes, this is another possibility, you can disable caching of specific entities, in this case taxonomy terms, by removing the cache keys, which are in the #cache property. It's easier to manipulate this array in PHP, but this should also be possible in Twig. You can try to get #cache with the filter |without('keys') and merge it later with first and second.

You can also try to remove #cache completely from the second rendering

{{ second|without('#cache') }}


because all relevant cache metadata should bubble up already from the first rendering of the taxonomy term.