Get results from "combined, either inner join"
Venn diagrams are not ideal to visualize join operations like a_horse commented:

Venn diagrams do not visualize joins, but set operations like union, intersect or except.

And his link to illustrate:

https://blog.jooq.org/2016/07/05/say-no-to-venn-diagrams-when-explaining-joins/

The twist, I think, is this: the Venn diagram is the right tool to visualize your objective, but the SQL JOIN is the wrong approach. You need a UNION (or equivalent) at the core of your query.
Title and the example query are misleading like that. What's more, you  start with:
SELECT * FROM ...

That gets all columns from all joined tables. Also not the expressed objective:

all organizations I'm either moderator or administrator.

UNION
The core query is:
SELECT organization FROM admin_organization     WHERE user = 10
UNION
SELECT organization FROM moderator_organization WHERE user = 10

UNION. Not JOIN. And not UNION ALL - we don't want duplicate organizations in the result.
The column name organization is also misleading (IMHO). Should really be something like organization_id for clarity.
The resulting set of unique IDs may already be all that's needed. To flesh it out with more (or all) attributes of the organization (columns of table organization), now we JOIN to the table:
SELECT *
FROM  (
   SELECT organization AS id FROM admin_organization WHERE user = 10
   UNION
   SELECT organization FROM moderator_organization WHERE user = 10
   ) x
JOIN   organization USING (id);

It's typically (substantially) cheaper to apply UNION on just the ID column, and then join. It may even be a necessity, if some of the columns have types have data types with no equality operator. More common than one might think. See:

How to sum the values ​of a json column filtered with regex?

Assuming referential integrity, enforced with FK constraints, nothing is lost in the join.
For convenience, it added a column alias in the subquery (organization AS id), so that we can use the simplified join condition with USING (id) which, in turn, allows us to use the simple (and now correct) SELECT * in the outer query to get all columns of the table organization without a duplicate ID column.
Alternative with EXISTS
Shorter equivalent that also avoids undue duplication or elimination of rows:
SELECT *
FROM   organization o
WHERE  EXISTS (SELECT FROM admin_organization     WHERE user = 10 AND organization = o.id)
   OR  EXISTS (SELECT FROM moderator_organization WHERE user = 10 AND organization = o.id);

Naming convention
A naming convention with descriptive names can avoid some of the confusion and noise. Use organization_id (or org_id if you prefer short names and there is no ambiguity) for all three: admin_organization.organization, moderator_organization.organization, and organization.id.