I am sure others can answer this more correctly from a facts perspective but as a fellow learner it helped me to more view it like this
In the first sentence what you are saying is “Because of A having happened, B” That is why it is past tense, if it was してから, it would be order of action, meaning “After I did A, B” which would also make grammatical sense here (I think) but it isn’t the intention of the sentence, you’re not looking to show the order of action of you ran and then got thirsty, you’re looking to express that your getting thirsty was caused BY the running you did in the past.
It may be helpful to remember から as “From” and expanding the meaning outward from there. So if its してから it’s “from doing A, b” this is order of action, but したから is “from having done A, B” This is like “because of A, B”
In the second sentence you’re using て form as “and” and you’re connecting the two clauses together.
I think the confusing part is that て can sometimes seemingly be saying something similar to から where it is expressing the cause/reason for B just like in this second sentence you provided. I am unsure if it is unable to rephrased as “買いたがったから, 週末も働いた. I don’t see why it’s not possible? But maybe it just sounds weirder like that and is less natural, or maybe its just simply not what the review sentence is looking for in this particular answer.
I think that starts to get into territory where someone else can better explain. I could be mistaken on some of this so others can feel free to correct me, but this is generally how I have been viewing it.