Tumblr user @beelmango devised a theory that Joseph Joestar is just a time lost Joseph Joestar between parts 2 and 3 and it seems a lot of fans agree. Here are some of their points:
• this is around the time he starts developing the beard
• cowboy hat, similar to what Joseph wears in part 3
• Mechanical arm
• Nobody has said that they aren’t the same character
In conclusion: same character, different timelines.
The comments that followed his post seem to only confirm the theory.
Episode 13 actually gives us the exact moment he comes up with his catchphrase.
It’s been right in front of us all this time!