Be specific, provide examples. Much of the things brains do that we call intelligent are totally unknown to us. We have, quite literally, no idea what algorithms the brain employs. If you want to talk about intelligence, I don’t know why you’re talking about neuron spiking. We don’t talk about semiconductor voltages when we talk about computer programs we’re working on.
AI systems are software, so if you want to build something brain like, you need to understand what the brain is actually like. And we don’t.
The brain-inspired papers have done realistic models of specific neurons, spiking, Hebbian learning, learning rates tied to neuron measurements, matched firing patterns, done temporal synchronization, hippocampus-like memory, and prediction-based synthesis for self-training.
Brain-like or brain-inspired appears to mean using techniques similar to the brain. They study the brain, develop models that match its machinery, implement them, and compare observed outputs of both. That, which is computational neuroscience, deserves to be called brain-like since it duplicates hypothesized, brain techniques with brain-like results.
Others take the principles or behavior of the above, figure out practical designs, and implement them. They have some attributes of the brain-like models or similar behavior but don't duplicate it. They could be called brain-inspired but we need to be careful. Folks could game the label by making things that have nothing to do with brain-like models or went very far away.
I prefer the be specific about what is brain-like or brain-inspired. Otherwise, just mention the technique (eg spiking NN) to let us focus on what's actually being done.