I do not wish to needlessly pay the short-term parking costs of a Uber or minicab driver. However, we also don't wish to pay the higher inflated cost of a “black cab” from a taxi rank. Therefore, is there somewhere easy to get to from the airport, where we can get picked up, ideally with a warm place to shelter in?
Manchester Airport really don't want people using cars at all for pick-up/drop-off; they charge taxis a lot of money to access the station, and they want to direct all private-hire to their "official partner" of Arrow Cars (who are really expensive relative to other private-hire or Uber, but cheaper than hailing a taxi). They've also tried repeatedly to make it more difficult for people to be given lifts by friends and family - all in the name of maximising income from their long-stay parking.
There are usually loopholes in their enforcement regime, but they get closed down over time (the Airport has to get legal authority to stop people doing something, so the loophole can stay open for a year or two until they get that).
Until recently (last year, I think), you could go to the train/tram station and get an Uber pickup there without any hassle, but they've (physically) restricted road access to that area, so you can't do that any more.
Last time I landed at Manchester Airport (which was this summer), the Uber driver arranged to pick me up at the departures drop-off area for my terminal. Note that they aren't allowed to wait there, so you have to be there waiting for them. I'm sure that the driver was breaking some Manchester Airport rule or other in doing this, so enforcement might be able to stop them, and not all drivers might be prepared to do this (I used the "call driver" feature in Uber to make arrangements).
As an alternative, you might try taking a local train to a nearby station (Heald Green is the closest, but I suggest East Didsbury because it has a waiting room) or even a tram, though there certainly aren't proper waiting areas at tram stops.
The train/tram station at Manchester Airport is integrated with the terminals (it's effectively a fourth terminal), so there's a long corridor with a travelator to the station, but it's no worse than changing terminals.
To be fair to the Airport authorities, the official reason they're hostile to cars is that the road capacity around the airport is at capacity, resulting in a lot of traffic jams, and they've been told that they can't expand the number of flights through the airport unless they encourage more passengers to use public transport.
Uber have a set of airport pick up guides, but Manchester is not on the list so I infer they are not authorised to pick up there https://www.uber.com/en-GB/airports/#Europe I normally get a fixed price quote when I get a mini cab from an airport, but haven't tried this specifically at Manchester.