There are two implicit parts to the question:
- what animals are as a taxon and
- why we call this taxon "animals" (= Animalia = Metazoa).
Organisms are classified mostly based on the understanding of the evolutionary relationships between them. Nowadays, evolutionary relationships for extant organisms are resolved using molecular data, in particular multiple gene sequences which provide many simple characters for phylogenetic reconstructions.
There are many branches on the tree of life (clades), each one has the potential to be treated as a taxon. To understand what clade deserves a name and corresponds to a particular rank, taxonomists search for well-supported clades with easily recognizable derived traits associated with them.
So, animals can be defined as the organisms belonging to the clade that includes organisms from Trichoplax and sponges to humans, or in other words: which includes the last common ancestor of these organisms and all its descendants. It is a well-supported deep clade and we call it Animalia = Metazoa based on the fact that most organisms that have been included in the typological definitions (i.e. definitions based on formal morphological inclusion criteria) of the traditional taxon "Animalia" starting from Linnaeus belong here. The typological taxon "Animalia" in turn is an improvement upon the earlier typological definitions of the word "animals/ζῷα/animalia" starting from Aristotle and both are attempts at formalizing the intuition languages originally have for the word "animals".
Thanks to shared ancestry, animals have a number of synapomorphies (features evolved in their last common ancestor) which include multicellularity, anisogamy, specific developmental signalling pathways, reduced mitochondrial genome and others. Synapomorphies are not inclusion criteria as in typology. Even if some other group developed a similar trait (e.g. multicellularity evolved several times among eukaryotes independently) or even if some animals lose some of these traits (although outside of zoological taxonomy, one extreme case might be transmissible cancers, such as canine transmissible venereal tumour which are essentially unicellular parasites recently derived from multicellular animals) those organisms do not become/stop being animals.
Oxford Dictionary in your citation does not define what an animal is. From the perspective of modern biology, it only mentions some traits found in some animals — not all animals have nervous system and not all rapidly respond to stimuli. Analogously, organisms that independently evolved fast responses do not become animals: they do not share ancestry with them and they lack their synapomorphies. Whatever similarities they might have with animals would be due to plesiomorphies (traits inherited from a more distant ancestors) or convergences/homoplasies (traits evolved independently).