If we take "UK's position" to mean the Political Declaration that accompanies the Withdrawal Agreement (both agreed by May with the EU, but 2-3 times rejected by MPs; the Declaration twice, the Agreement thrice) then this [tentatively] common vision with the EU says:
[23.] The economic partnership should ensure no tariffs, fees, charges or quantitative restrictions across all sectors, with ambitious customs arrangements that, in line with the Parties' objectives and principles above, build and improve on the single customs territory provided for in the Withdrawal Agreement which obviates the need for checks on rules of origin.
[29.] The Parties should conclude ambitious, comprehensive and balanced arrangements on trade in services and investment in services and non-services sectors, respecting each Party's right to regulate. The Parties should aim to deliver a level of liberalisation in trade in services well beyond the Parties’ World Trade Organization (WTO) commitments and building on recent Union Free Trade Agreements (FTAs).
[30.] In line with Article V of the General Agreement on Trade in Services, the Parties should aim at substantial sectoral coverage, covering all modes of supply and providing for the absence of substantially all discrimination in the covered sectors, with exceptions and limitations as appropriate. The arrangements should therefore cover sectors including professional and business services, telecommunications services, courier and postal services, distribution services, environmental services, financial services, transport services and other services of mutual interest.
So they aim for a fairly comprehensive deal in both goods and services. That's perhaps even more ambitious than the deal with Japan or Canada, so call it Canada+ or something like that. Do note that in the jungle of Brexit proposals, the same term has been used to mean something somewhat different
Canada plus plus would mean a hard Brexit and an arms-length relationship with the EU based on reducing tariffs and trade barriers and harmonising regulations. It is attractive to the Brexiters because it would keep Britain out of the single market and customs union. But many Remainers would consider it to be very similar to a no-deal Brexit. It does have the advantage, however, of being supported by the dissenters in the House of Commons – as long as the backstop is not part of the deal.
The take-home message from this is that the future relation can be reached either by upgrading a no-deal or downgrading from May's Withdrawal Agreement. The May-EU solution is the latter. Those who can't stand the backstop want the former... or they want something completely different (zero tariffs etc.)
So I'm largely eschewing what "hard brexit"/"no-deal" means in terms of future relations, since it depends on who's talking and what else they envisage should happen after the event. If nothing else happens, a no-deal Brexit just reverts the UK to WTO rules, in terms of trade. Even this is not a simple affair, because the UK is currently [re]negotiating its (future) WTO schedules.