These are not even remotely similar, despite the name. Things are moving very fast, and the sort of chat-based interface that you describe in your article is already obsolete.
Claude is the LLM model. Claude Code is a combination of internal tools for the agent to track its goals, current state, priorities, etc., and a looped mechanism for keeping it on track, focused, and debugging its own actions. With the proper subagents it can keep its context from being poisoned from false starts, and its built-in todo system keeps it on task.
Really, try it out and see for yourself. It doesn't work magic out of the box, and absolutely needs some hand-holding to get it to work well, but that's only because it is so new. The next generation of tooling will have these subagent definitions auto selected and included in context so you can hit the ground running.
We are already starting to see a flood of software coming out with very few active coders on the team, as you can see on the HN front page. I say "very few active coders" not "no programmers" because using Claude Code effectively still requires domain expertise as we work out the bugs in agent orchestration. But once that is done, there aren't any obvious remaining stumbling blocks to a PM running a no-coder, all-AI product team.