I played through it again, and this time when I tried to nickname it, I ended up nicknaming my Farfetched instead. It even asked if I wanted to nickname the Bulbasuar I just got. The difference was last time, I had a few Pokemon caught and in my party, while this play-through, Farfetched was the only Pokemon I had when I received Bulbasuar. Maybe the easiest way to fix this problem is to put the name rater in Visade city, not give the player a chance to nickname the Bulbasuar, and have Professer Roy mention the name rater instead.
What I mean is, when I used Cut in the field, my character's look momentarily turned back into the original character. Check the attachment for a picture about what I mean.
Also, I didn't say this in the first post but I think it's cool that there are some Pokemon you really have to search for. I remember I ran into a Nidoran(M) and accidentally killed it, so I ran around for about 15 minutes trying to find it again (since I really like Nidoking) when I realized that when you get an updated version of this past the second city, I'd have to restart and I'd lose the Nidoran anyways. XD I felt stupid.