⊕ is a standard symbol for this kind of math. The symbol itself is ancient because it's so simple, so I don't see what Bayer's aspirin logo has to do with it.
It acts as a normal +, mostly. When you're dealing with modulo math, the "normal" plus becomes a bit weird as there are rules attached to a number expressed as "(a + b) mod c", so mathematicians often use symbols like ⊕ to mean something like "+, but different". The second link you posted does the same, it acts sort of like normal addition, conceptually, except it's not done on actual numbers but groups.
In definitions like these, you may as well use a peace symbol or a picture of a frog; "some operation ⊕" means "there is some operation we write down like this, and it does this and that".
Another place you may find ⊕ is when it's used to represent XOR in some cases; (a + b) mod 2 is a bitwise XOR when operating on single bits (again, it means "normal addition except with weird rules", namely the mod 2 that makes you throw out anything larger than the last bit).
It acts as a normal +, mostly. When you're dealing with modulo math, the "normal" plus becomes a bit weird as there are rules attached to a number expressed as "(a + b) mod c", so mathematicians often use symbols like ⊕ to mean something like "+, but different". The second link you posted does the same, it acts sort of like normal addition, conceptually, except it's not done on actual numbers but groups.
In definitions like these, you may as well use a peace symbol or a picture of a frog; "some operation ⊕" means "there is some operation we write down like this, and it does this and that".
Another place you may find ⊕ is when it's used to represent XOR in some cases; (a + b) mod 2 is a bitwise XOR when operating on single bits (again, it means "normal addition except with weird rules", namely the mod 2 that makes you throw out anything larger than the last bit).