We derive novel concentration inequalities for the operator norm of the sum of self-adjoint operators that do not explicitly depend on the underlying dimension of the operator, but rather an intrinsic notion of it. Our analysis leads to tighter results (in terms of constants) and simplified proofs. Our results unify the current intrinsic-dimension and ambient-dimension inequalities under independence, strictly improving both categories of bounds (such as by Tropp and Minsker). We present a general master theorem that we instantiate to obtain specific sub-Gaussian, Hoeffding, Bernstein, Bennett, and sub-exponential type inequalities. We also establish widely applicable concentration bounds under martingale dependence that provide tighter control than existing results.