LOL, nicknames can stem from far in the past when a place was far different. NJ is the Garden State. See many gardens in NJ these days?
In any case, if you stick with pre-med, go with the vibe that you like the best. The name of your undergraduate school won’t mean much. However, there also isn’t much differentiation of the pre-med curriculum at these schools, so I would not consider that a factor.
Also, if you decide against pre-med (or can’t get in), where you go for undergrad may matter more.
JHU has a reputation for being very research-oriented, BTW.