You're correct that there would be no “backstop” if there is no deal whatsoever. As you rightly put it in your edit, the backstop is for the scenario where the EU and UK agree on a withdrawal agreement but fail to find any other long-term solution for the Irish border in subsequent negotiations.
Beyond that, understanding what the whole backstop business is about requires looking in details at the mechanics of the negotiations themselves.
What the EU has been pushing from the start (and the UK was forced to agree in principle in the summer of 2017) is that an orderly Brexit would entail two separate agreements:
- A legally binding withdrawal agreement that must be signed and agreed by all parties by March 2019. The EU wants that deal to include provisions on a payment (for what the EU regards as the UK's current commitments under the EU 7-year budget plan running from 2013 to 2020), long-term guarantees for EU citizens currently living in the UK and a solution for the Republic of Ireland-UK border. The most difficult issue presently seems to be the border question.
- A “future relationship" agreement covering trade, access to the single market and all sorts of ancillary agreements (on things ranging from airlines operating rights to scientific research grants). That's much more complicated and very difficult to finalize within two years (let alone the mere months available now).
There have also been many discussions on some sort of “transition period” bridging the two, with many competing interpretations of what that would exactly entail.
Initially the UK was keen on tying all these issues together and the EU insisted on sequential negotiations, discussing a limited withdrawal agreement first and then moving on to the future relationship between the UK and the EU. The UK yielded in the summer of 2017 and the EU then demanded that “sufficient progress” be made on the first agreement before opening discussions on the second one.
Now, Ireland and the EU want the border issue to be dealt with in the first agreement but what arrangements could be put in place in practice also depend on the kind of relationship the UK and EU will ultimately agree on. That's where the backstop comes in: the first agreement would leave open the possibility that some other solution is found during the negotiations on the second agreement but would already include a (rather unpalatable, to the UK at least) solution if an alternative is not forthcoming.
What this achieve is to protect Ireland's interests ahead of the negotiation on the second agreement and to extend some of the leverage the EU currently has into the negotiations happening after March 2019 (“You don't like the backstop? Come up with something we like, or else…”). It would also provide a kind of fail safe mechanism on this issue if the negotiations on the second agreement falter (including as a result of an election and cabinet reshuffle).
So in December 2017, a broad outline of what the first agreement might look like was agreed and it included the “backstop”. Legally, it will only come into force if that outline is translated into an actual agreement and both parties agree to it. If there is no deal whatsoever, then the UK is not bound by this either, legally speaking.
Based on this, the EU has accepted to start discussions on the future relationship but nobody expects those to be concluded in time for Brexit. They also insist on the December 2017 principles being implemented no matter what, even as the rest of the negotiations proceed, and published a draft agreement to that effect. By contrast, some British officials (most notably David Davis when he was still Secretary of State for Exiting the European Union) suggested that everything was still open and that the UK could backtrack on its December 2017 commitments if it didn't get enough concessions from the EU on the issues the UK regards as essential.
Importantly, even if the UK actually leaves with no deal and both sides take some unilateral measures to mitigate the negative consequences, they will still need to negotiate various agreements on air transport, financial services, medicinal products, etc. and eventually some comprehensive trade agreement. And the EU already announced that its starting position in any subsequent negotiations would be based on its current goals regarding budget contributions (the “divorce bill”), citizens' rights, and the Irish border. So the backstop wouldn't automatically be off the table.