You can use the following basic query setup to get the types of conversions being used by your LinkedIn campaigns.
Dimensions: Conversion Type
Example query in Google Sheets:
Warning! Many other metrics are not compatible with the Conversion-specific dimension so you won't be able to query them at the same time (they may return an error or will be empty values). Due to how the API works, you also cannot request other dimensions, so you can't even get a join key from that. Sadly, as this is an API limitation, there's nothing we can do about this.