Postgresql – Recursivly get a Tree Via self joined table

Using other questions here & Postgresql documentation I've managed to build a many-to-many self joined table.

However adding a WHERE clause is giving me trouble.

Problem:

A Category can have many child categories, and many parent categories. Given a category.Id, I want to retrieve the category, the category children, childrens children and so on.

Example: given this structure:

child_1
    child_11
        child_111
        child_112
            child_1121
    child_21
child_2

Given: a clause of id = child_11

Expected results:
child_11, child_111, child_112, child_1121,

Actual results: child_11, child_111, child_112

Here is my attempt: http://sqlfiddle.com/#!17/3640f/2

In case Sqlfiddle is down: https://www.db-fiddle.com/#&togetherjs=LhDjxfPHo6

Note: I don't care about duplicating the where clause, my application can handle that

Table structure:

CREATE TABLE Category(id SERIAL PRIMARY KEY, name VARCHAR(255));
CREATE TABLE Categories(parent_id INTEGER, child_id INTEGER, PRIMARY KEY(parent_id, child_id));

ALTER TABLE Categories ADD FOREIGN KEY (parent_id) REFERENCES category (id);
ALTER TABLE Categories ADD FOREIGN KEY (child_id) REFERENCES category (id);

Table data:

INSERT INTO Category(id, name) VALUES (1, 'parent_1');
INSERT INTO Category(id, name) VALUES (2, 'child_1');
INSERT INTO Category(id, name) VALUES (3, 'child_2');
INSERT INTO Category(id, name) VALUES (4, 'child_3');
INSERT INTO Category(id, name) VALUES (5, 'child_1_1');
INSERT INTO Category(id, name) VALUES (6, 'child_1_2');
INSERT INTO Category(id, name) VALUES (7, 'child_1_1_1');
INSERT INTO Category(id, name) VALUES (10, 'child_of_many');
INSERT INTO Category(id, name) VALUES (11, 'parent_1');
INSERT INTO Category(id, name) VALUES (12, 'parent_2');

INSERT INTO Categories(parent_id, child_id) VALUES (1, 2);
INSERT INTO Categories(parent_id, child_id) VALUES (1, 3);
INSERT INTO Categories(parent_id, child_id) VALUES (1, 4);
INSERT INTO Categories(parent_id, child_id) VALUES (2, 5);
INSERT INTO Categories(parent_id, child_id) VALUES (2, 6);
INSERT INTO Categories(parent_id, child_id) VALUES (5, 7);

My query which is giving me the children, but not the childrens children etc.
If I remove the WHERE clause's I can get all rows:

WITH RECURSIVE categories_category AS (
  SELECT id, 'Category' AS COLUMN_TYPE, c1.name 
  FROM Category c1
  WHERE c1.id=2
UNION
  SELECT c2.id, 'Category' AS COLUMN_TYPE, c2.name
   FROM Category c1
   INNER JOIN categories cs1 ON c1.id = cs1.parent_id
   INNER JOIN Category c2 ON c2.id = cs1.child_id
   WHERE cs1.parent_id = 2
) SELECT * FROM categories_category

Edit: More Detailed example:

Given the following category row's, I'd like to be able to run a query given a WHERE clause the matches the id of warmStoreAlcohol and get a result of:

+---+------------------+
|id |name              |
+---+------------------+
|2  |warmStoredAlcohol |
|3  |vodka             |
|4  |beer              |
|5  |frozenBeer        |
+---+------------------+

coldStoredAlcohol would give a result of:

+---+------------------+
|id |name              |
+---+------------------+
|6  |coldStoredAlcohol |
|5  |frozenBeer        |
|7  |cooler            |
+---+------------------+

The database structure will not change often. In this example 'frozenBeer' has two parents, and should return for querying both warmStoredAlcohol and coldStoredAlcohol.

I am open to changing the table structure, adding new tables and even upgrading postgres version etc. The database will hold ~2,000 rows, therefore I value an easy-to-understand table structure over a super complicated optimal one. (But any solution is better than my broken one)