Does billing clients through a numbered company instead of my own name change whether I'm a personal services business?

No, billing through a numbered company instead of a named corporation is purely a naming choice and has no bearing on whether the underlying arrangement meets the personal services business test. CRA's analysis focuses entirely on the substance of the working relationship between you and your client, control, ownership of tools, financial risk, and integration into the client's operations, none of which is affected in the slightest by whether your corporation is called "123456 Ontario Inc." or has a chosen business name.

This comes up because some people assume a more anonymous or generic-sounding corporate name somehow distances the arrangement from an obvious employment relationship, but CRA looks straight through the name to the facts of how the work is actually performed and structured, exactly as it would for a corporation with a client-facing brand name. A numbered company that is, in substance, an incorporated employee working for one client under close supervision is treated no differently than a named corporation in the identical situation.

If you're concerned about PSB risk, the naming convention for your corporation is simply not a lever worth spending time on, the working relationship's real substance, discussed elsewhere, is what actually determines the outcome.