Arts and Crafts Service Company:
Examples of services type companies are doctors, accountants, architects, actuaries and lawyers. I can only think of one type of art or craft business that would fall under this classification. And that would be an arts or craft designer who comes up with the designs for other related businesses but doesn't make any product for resale.
One example of this can be a fabric designer. Fashion designers come to my business looking for specific surface design of fabric for their garments. I come up with the pattern, design and color scheme and use software to replicate the design in a shareable image file that the designer can send to their fabric dyers. I am paid for my design work but I am not involved in the manufacturing process.
If you only make design prototypes, that type of craft business would also fall into the service category. One example - a jewelry designer that designs and makes a sample piece of jewelry based on the customer's - maybe a jewelry manufacturer - specs. In essence, these types of arts or craft businesses are consultants.
One major tipoff that you're a service type of craft company is if you don't have any appreciable inventory. Most service type companies only make purchases for the job at hand so they won't carry an inventory - purchases will be expensed. If they do retain some purchases, the amount is inconsequential especially when compared to a merchandising or manufacturing company.
Arts and Crafts Merchandising Companies:
These are retail businesses such as a gallery, craft store, online shop or boutique. A merchandiser purchases goods from an art or craft business and in turn sells the goods to the end user - a consumer like you or me. In many cases, arts and crafts businesses are both merchandising and manufacturing companies. You handcraft your products and sell them yourself either online, at shows or in a storefront.
To me it is the best of both worlds if an artist or crafter can generate enough business to have their own retail location to sell their products. I have been in storefronts and galleries where a portion of the shop was the artist's studio. While this is too much of a distraction for me and somewhat of a potential lawsuit waiting to happen depending on the type of tools and chemicals in use, it is a great marketing tool.
Arts and Crafts Manufacturing Companies:
This type of business makes the tangible arts and crafts products that are sold to either merchandisers or directly to the customer. Going back to the service company jewelry designer, after creating the prototype, instead of selling the design to another manufacturer, the jewelry designer creates multiple copies of the piece of jewelry and sells the jewelry to either merchandisers or the consumer.
As you can tell, it's possible for you to wear different company type hats as an owner of an arts or crafts business. If you make and sell your product directly to the customer you are both a merchandiser and a manufacturer. If you make your product and sell it to a merchandiser you're a manufacturer only. Designers who only sell the concept have a service type business.