Voting for "simply casual speech" (obviously ;)). Here is one link that shows how using "タメ語" sometimes to seniors can better communication. IMO the writer here is using it to simply mean "casual speech".
Online definitions such as these also tend to focus on the description of the nature of the speech.
However, the etymology of this term is "speech between equals" so I'd think some would find the word loaded.
The word itself is slangy and comes from gambling, and was traditionally used among gangs etc. So if you'd use to a nice lady or teachers, or company bosses, it would sound weird and rude because it's kinda like saying "what the hell" or something to such groups. IMO it's not necessarily rude because it implies that the other party is equal (which I think the word doesn't).
For example, I might ask a senior person タメ語でいいですか？ to become more friendly. If you are sufficiently familiar with the person and the person is "within range" you could use タメ語, I find this perfectly ok. Or you could use タメ語でお願いします if a senior person is using 敬語 to you. This can even indicate more respect, because you are asking the other guy not to use 敬語. Because it's a slang though, you have to be a bit careful to whom you can use it. E.g. teenager to 20 something-ish senior in バイト setting should be fine.
タメ語 switching is an extremely delicate matter, so I won't recommend anyone to try it too easily. Although if it's clear you are not a native, people would be less likely to be offended when you make wrong moves.