I'm guessing that most centrists would accept the moderate definition above, that they occupy the 'centre-ground' of politics being neither left nor right. Having said that its clear to me from comments I've seen by avowed centrists on Twitter that some centrists see themselves as progressives - people who want reform, who want to 'improve' things. And some clearly even see themselves as centre-left.
Where does all this get us? What is the political centre? One of the most interesting commentators on these issues is the neurolinguist George Lakoff. For Lakoff, there is no political centre, essentially you can only be left (progressive) or right. He bases his views on the idea that conservatives follow the 'strict father' model and progressives the 'nurturing family' model*, and that this is the base of their world view and political views. This extract from an interview in the Guardian explains how he thinks the left should approach politics and his view of moderates and the centre ground. It's worth reading the whole article:
"This is what he believes it would take to refashion the progressive [left] mindset: the abandonment of argument by evidence in favour of argument by moral cause; the unswerving and unembarrassed articulation of what those morals are; the acceptance that there is no "middle" or third way, no such thing as a moderate (people can hold divergent views, conservative on some things, progressive on others – but they are not moderates, they are "biconceptual"); and the understanding that conservatives are not evil, unintelligent, cynical or grasping. Rather, they act according to the moral case as they see it." [my italics and brackets]Incidentally, I agree with him. The left spends too much time and energy on evidence-based political argument, which just doesn't cut it with many voters, and too little time on the moral case for change.
I would go further than Lakoff. As far as I can see 'biconceptual" individuals - who think of themselves as being in the political centre - are conservatives. Maybe with a small 'c' but still conservatives and on the right of politics. Why? Because they tend to be people who are more comfortable with the status quo than people on the left could ever be. Because, for example, they accepted the New Labour attack on benefits claimants and the Iraq war, they accepted the 'need' for austerity cuts and are generally comfortable with privatisation and the neoliberal agenda. Nobody who is genuinely on the left could agree with any of that.
Of course, the term 'moderate' is a boon to the mainstream media. For them, a moderate is a sensible person who accepts necessary 'reforms' including cuts to public services - as I said earlier someone who accepts the status quo. Thus the epitome of centrist politics, The Guardian, praises and promotes sensible, moderate politicians such as Yvette Cooper who is currently the pin-up politician of English centrists.
For me, what provides conclusive proof that centrists are conservatives are the relentless attacks they make on Jeremy Corbyn, a decent man, a democrat, who leads a Labour Party, which has a modest Labour manifesto for social democratic reform in the UK. What really troubles conservatives of all stripes about Corbyn is not that he is a Marxist (they know he isn't) but that he is the genuine article, someone who would actually implement the manifesto and bring about real change and break the neoliberal stranglehold on the UK.
*there isn't scope in this post to discuss Lakoff's ideas in more detail but you might want to read his book The Political Mind to find out more. You won't be disappointed.