If I share a remaster that has only subtle differences from the original distro (such as removing nonessential software, installing different internet connection software, or unlocking the root account if the designer locked it) rather than a more obvious major change (such as a different desktop manager), would I still have to give it a new name? Do designers rename variants because copyright law requires them to, or do they just like giving their variants new names?

In most cases the artwork and names of the distro's have some legal protection. For example, you can't call your distro Ubuntu and offer it. If you are talking about some friends then don't worry. The RH company went berzerk maybe 10 years ago and went after clones that used their artwork and even in the embedded documentation that said the RH name anywhere in the offering.

You have to check how distro's are offered, not all tools and apps can be legally offered. One of the rules is that you have to offer sources. That doesn't mean you have to have them on the disk, just that you have them available some place.

You can easily use one of the creator type deals to get past it all. SuseStudio can create what you want. You can share it on that site with others.

As jefro stated, you have to check for any distro that you want to respin. For example, the license for Slackware does not allow to use the name Slackware for a re-distribution that contains any change without permission from Patrick Volkerding. Other distributions handle that different.

I assume that includes using the name with a modifier, e.g., "Slackware Variant #653426653478"?

AFAIK, yes.

BTW, I was just looking at a different page and found this, someone might get a laugh out of it-

http://www.overclockers.com.au/image...27apr12/14.jpg

Next up, 'Slackware Vanilla Cream Soda'

Since Slackware is a registered trademark, yes, that is included.

Nope. Slackware would include every single flavour of soda you can imagine, spread out over 7 CDs.

the software is fine as long as you fallow the licenses
GPL2,GPL3 ,MIT,cc-by-... , on and on

-- the 800LB Gorilla in the room --
-- The Artwork --

get permission for ALL and I do mean ALL the artwork
or
make your own