Ok, I think I've figured it out. Basically, I could most likely keep my current name because, 1) The other group is in a totally different industry, and 2) The name is not trademarked. I think I just might change it though to "Glow-in-the-Dark Games"

At some point, when your company reaches a certain level of profitability and/or visibility, you'll want to re-address this, with your lawyer.

You mean, just to make sure I'm doing everything correctly?

What he means is something like this Mojang vs Bethesda
Even if this seems to be on an other 'level', once you gain some awereness someone might think that it is a clever idea to fill in a lawsuit, to protect their own property or to make some money out of it. Even if you might win in the end, are you willing to invest much money with the risk of loosing ? What if you loose, will it destroy your dream ? What is more important to you, to use this company name or to protect your company from potential dangers ?