We may be getting twisted up between legal and mathematical definitions here, but I do not see the outcome of 2D6 as random, but rather as the product of two separate random events. The quintessential randomness is in the roll of each singular die, not in any output you compute by combining several die rolls. I would however say that that output is determined "by chance".
Like I say, have no idea about how the meaning of those words would be adjudicated in a court of law by any given judge, but that is the plain meaning of those words to me.
Welcome to the legal field of statutory interpretation, in which we try to give precise meanings to imprecise words according to a set of vague principles!
There's no question that you are drawing a real distinction. A uniform probability distribution is meaningfully different than one that is not. See technical note below. But then the trick is to figure out how that distinction maps onto our language, and that's not easy because of those words at the end of your post. "To me." Yeah, we're all stuck in that epistemic quandary to some extent.
Brief anecdote: in law school, I took a class taught by John Manning (who, rather improbably, became the Dean of the law school, and then of the whole university. It's not common that Jewishness is an asset on a resume but in this case . . . ). He and I used to spar all the time. He's a textualist; that is, he's a big fan of using only words to discern meaning. And in class, he would say things like, "and this, after all, is what this word means." And I would say "but couldn't you equally argue that the word means this other thing." His response would be, "that's clever, but isn't it really this?" Of course, that raises the question of determining what's clever and what's true. As many students in the class came to observe, that's a question of authority as much as one of language. Interpreting words can be tricky.
Anyway, back to randomness. To get around the "to me" problem, we often try to look at language how it's actually used. For instance:
1. A guy comes up to you in the grocery store parking lot. He says, "I'm starting a business. If you loan me $100, I'll pay you back $200 next year." I think most people would think it appropriate to respond, "No offense, but I don't loan money to random people." Do you? But of course, on your definition of random as uniform probability distribution, it's not random at all. You only ran into this guy because you shopped at this store and not one in a different city or even across town; because you allowed him to approach you instead of walking away briskly; because you go to the store at all, etc.
2. The victims who were killed by the bullets meant for Trump in Butler PA. We call those deaths random, right? But they weren't random at all, per your definition. For one thing, you had to have been at a Trump rally to be shot. Second, you had to in the line of fire from the specific place where a gunman could get off a shot at Trump. Most people at the event had a 0% chance of being shot.
3. When people are asked to name a random number between 1 and 100, the results are not uniform. For one thing, almost nobody guesses 2, nor multiples of 10. I watched a video about this. I think the most chosen numbers (by a considerable margin, even) are 37 and 73. There's definitely a gradient.
But if I ask you to name a number between 1 and 100, any number, I think we'd describe that as random. And if I set the humidity in a commercial greenhouse to that number, my business partner might be justifiably outraged that I just set the growing conditions to a random number pulled out of someone's butt. Saying, "well, it's more likely to be 73 than 2" would not answer the criticism.
In general, I think that people use the word randomly to denote a process in which the outcome is not fully predictable, or if an outcome (or an input) is established by reasons that are not apparent.
Technical note: describing the probability distribution of a 2d6 isn't fully explained as "two separate random events." I could weight a 12 sided die so that its probability distribution would be the same as 2d6. You could say, well that simulates the 2d6 probability, which is true in a sense; but it's also true that the 2d6 actualizes a type of probability distribution that can be actualized in other ways as well. But that's not really important and also I might not be right about this point and I don't want to argue it.