The colleague may have suggested the soft play but it was the OP that brought up that they were then going out for a meal. There is quite a price difference between going to a soft play and soft play plus a meal. Soft play here is ~£6 and £3.50 for a coffee but the local pub is £15 for a sandwich and chips, £6-10 for a childs meal plus £5 for a coke and fruit shoot. That takes a morning out from £10 to £40.
The colleagues child then got upset as they also wanted to go out for a meal and when Abi was getting upset about this the OP said maybe next time. After this Abi felt unable to not say no to the her child.
Now this is on Abi, it is not the OPs fault that Abi couldn't afford /want to go for the meal too but it also wasn't the OPs place to say next time they could go for a meal. I can see how this could annoy Abi - and how as I said "rightly or wrongly" Abi had a problem with it. Personally, I would say that Abi is wrong to have assigned the blame of dangling the "treat" of dinner out as the OPs fault and it was on Abi to explain that they can't do it today or even at all. I wouldn't have mentioned the meal in ear shot of the children as children always want to continue the fun but at the same time if I was Abi, I would see it as my responsibility to manage my child's expectations and explain to them why we weren't going for lunch and why OP was.
The next playdate, following Abi mentioning the last trip "skinting her" and then the OP making another promise to Abis child that money will be spent for them to get something. When children talk about any mythical being bringing presents the unwritten code to other parents is "we or nearest family will be getting presents under the guise of the birthday fairy" to use this language on the back drop of the meal and pub date I can can see how again "rightly or wrongly" Abi has got annoyed. I would see that my role as parent in this example would be interject and remind them that they would get plenty of presents for their birthday or have lovely toys at home..
It's possible to see how another person came to a position even if you don't agree with that thought process. Based on the OPs representation of her interactions with Abi, Abi finds it difficult to communicate when things are beyond her budget especially when it's infront of her child, I wonder whether she didn't want you to think that she couldn't afford to join you for the meal too and struggled with finding the words to communicate that she couldn't come to the pub too even though she hadn't been invited to come along.
Family finances are tight for some families right now, is she full time or part time? She might have a higher mortgage, the new car that comes with hefty monthly costs, the daily coffee is an expensive habit. All together it could be a lifestyle that doesn't have much give. There could be hundred and one other reasons why she doesn't have the means for extras beyond what she has suggested.
In OPs position, I'd let Abi know that she didn't mean any harm, she didn't intend for them to join them at the pub or have to go and buy another present. That the the children enjoy playing together and it's ok to stick to whatever pre-agreed activity is
I have a friend, who I now know is on a tight budget but one of the first afternoons out we got ice creams/lollies for the children. Hers had two bites of the ice cream and didn't want the rest. She couldn't eat the rest as she can't eat dairy nor can my daughter so it went in the bin. As she did she said "that was an expensive mistake". That has framed things I plan with her knowing, that for her an ice cream from van is an expensive treat and adjusted things my child gets accordingly so it's means that they both roughly get the same. Play dates with this family are often places where we can bring our own lunches and there aren't too many upfront costs.