I love that the OP put this out there as a learning experience. Regardless of the fact that he has an MBA and a technical background, he cheerfully admits he was wrong and tries to do a better job managing expectations in the future. I'll provide a blow-by-blow analysis.

>>“We need to under promise and over deliver.”

Great! But, why? There has to be a concrete, preferably measurable, reason(s) to have this mindset. As other commentators noted, what he did wasn't really under-promising, but simply adhering to the original schedule, which implies that the customer was on-board with the agreed timeline.

>>The project moved along and thanks to some long days we actually managed to deliver functionality ahead of schedule.

What did he expect to get out of this? What did he tell the team to justify this?

>>We continued our iterations, tuning functionality and consistently over delivering when the team started to become disgruntled and exhausted.

Even with the right motivation, burnout is a constant issue for services staff in any industry, especially when the engagement lead doesn't monitor both sides (customer and internal). He would have monitored this more closely, or would not even have gone down that path if his internal team were billing him on an hourly basis, as it would have forced him to more concretely justify to himself and his partners what he expected to get out the self-imposed accelerated schedule.

>>To continue to over deliver, the team had been working very long hours, weekends and some had missed important family appointments.

Live and learn. Seems like you are somewhat charismatic, or have a hold on your employees that made them put in the long hours and miss family time without pushing back on you.

>>The next day we presented to the client, they were surprised at what we’d achieve in such a short time and were happy that we were ahead of schedule.

As engagement manager, you should have ensured that this wasn't a surprise to the customer! It would have allowed you to more quickly gauge whether (a) it's worth continuing on the death march and (b) how the customer will react.

>>The discussion turned to the next set of deliverables, the effort and the expected delivery dates. This is where it all started to fall apart.

No -- we need to be very clear here. Things started to fall apart well before this due to: (i) lack of identifying value exchange for the death march on the first deliverable; (ii) lack of continual communication with the customer.

>>“Why can’t you do it? We’re not asking for anything more, it’s actually less than what you delivered before.”

The standard answer here is:

(i) We can't do it because the first deliverable was a one-off to help you reduce time-to-market / internal target date, but this second deliverable will require additional staff that I don't have. From a resourcing perspective, I'm not going to charge you for the additional hours that my staff spent on the accelerated schedule for the first deliverable because I made that decision unilaterally. We must stick to the same duration for the second deliverable.

(ii) I know you're not asking for anything more; that's why we're able to commit to our originally agreed upon deliverable duration (e.g. 2 weeks), and nothing shorter.

(iii) If you want us to work weekends, as per the Statement Of Work ("SOW") that you signed, you'll need to pay for the additional weekend and holiday staff shifts for the second deliverable [context: sounded like the OP did in fact have a contract in play, and hopefully had language in the SOW that covered increase in base rates for weekend & holidays]. We'll swallow the increase in costs for the first deliverable because we did it of our own volition.

(iv) I'm glad that we confirm our original agreement regarding the nature, specificity and scope of the second deliverable (i.e. that it is "less than what you delivered before"). This further emphasizes that our originally agreed upon duration for the second deliverable is firm and we can deliver with the agreed upon duration.

>>Over the next few meetings, with significant effort, the discussions started to turn around. I had been reborn and now realised:

There's hope for you yet! In most situations, this is where the customer escalates, goes over your head and tries to get you off the project, or where the firm partner/general manager tries puts a PIP on your record because your customer relationship management skills (the number one reason you're typically chosen to become an engagement manager) was so severely lacking. Seriously, that was a great turnaround if the OP did that himself.

>>It’s not about the effort, it’s about the outcome. The client did not care how much effort we’d put in.

In general, this is true. However, as part of your job you are supposed to tie in the effort with the outcome, whether via narrative, task-based estimate or simply pure dollar terms. Do this, and your job becomes easier explaining timelines and pricing deliverables.

>>We needed to become a partner

Yep. However, as part of your job, you must be cognizant when a psychopathic or aggressive customer has no interest in becoming a partner and only extracting as much value as possible out of the current deal. Customers can flip, and engagement managers must always be taking the customer's temperature to protect the deal, the team, and the deal margin.

>>The expectations had to be reset

As others noted in this thread, they weren't managed properly. OP was fine up until the kick-off when the baseline timeline was established and agreed upon, but then went off plan for no compelling reason(s).

>>As painful as this was, it did teach me a lot. Get on top of this before it blows out.

Amen.

>>Under promising and over delivering is for suckers.

This last sentence is wrong. Forget about under promising. Just focus on the over-delivering part. Only over-deliver if you have the appropriate risk appetite, enough deal margin you're willing to sacrifice, and compelling exchange of value. Examples of exchanges of value are:

(i) "If we make a great impression that is sustainable in the long run, we can leverage this influential customer for references in the future!" --> Make sure your sales team has the balls to follow-through on this. Surprisingly, many organizations and field teams have a hard time asking for references for future deals.

(ii) "When we were talking, the customer kept referring to feature X that's gating the next phase of their program and holding up future deals. If we deliver a proof of concept showing that we can deliver this functionality, it might clear the obstacle for future services deals." --> Make sure your product/engineering group can deliver, and that your support group will actually support the new 'feature' that you deliver.

(iii) "The customer is asking us to bring in the schedule. We've got no other deals on the table and all of our guys would be on the bench (i.e. idle) otherwise. Let's swallow the cost for a short period of time." --> Engagement manager needs to go into 'expectation management' overdrive and find a narrative that works for both parties.