"By 'abstract labor' Marx means work done simply in order to earn a wage, rather than for the worker's own specific purposes. Thus making a pair of shoes because one wants a pair of shoes is not abstract labor; making a pair of shoes because that happens to be a way of getting money is."
"Marx: A Very Short Introduction"
By Peter Singer
Pg. 33
Abstract labour is labour when thought of independently of the labourer's individual being, but as the thing which can produce value. Perhaps an example would be: "I have 100 tree trunks. I want to turn them into planks. I need to hire labour to operate the machines that do this."
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You make it seem like the "abstract" in "abstract labour" is a purely mental abstraction, i.e. a nominal abstraction.
It isn't: it's a real abstraction. In the act of exchange, through the medium of money, various private acts of concrete labour expenditure are made commensurable. An abstraction is made from the sensuous particularity of shoe-making, automobile construction, hamburger flipping, etc.
Now, having accepted that, you still have a lot of disagreement between Marx philologists about where value "comes into existence", in the production process or in the act of exchange. I personally think these debates miss the point, since the point isn't a productive vs. circulative theory of value, but rather that since all production in capitalism is production for exchange, the logic of exchange exerts an influence upon the structuring of the production process itself, which is why Marx examines it in Vol. I from both its material aspect and as a process of valorization.
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I think that the example is probably correct (this probably is abstract labour) but misleading (it does not explain why it is abstract labour). This example would be better to illustrate alienated labour. Abstract labour is labour when thought of independently of the labourer's individual being, but as the thing which can produce value. Perhaps an example would be: "I have 100 tree trunks. I want to turn them into planks. I need to hire labour to operate the machines that do this." Here I am talking about abstract labour. It's got nothing to do with any individual, or how they experience the work. To realise my need for abstract labour, I go out into the labour market and buy the best value (i.e. cheapest per output) labour power (comodified labour) I can.
So all in all, and without knowing the context, I think the description misses the point a little. It's more about the alienation of labour than the abstraction of labour. Though you can't really have the one without the other, but both concepts, whilst co-dependent, are distinguishable.
(p.s. my marxist jargon ain't perfect, so can others please correct me if I'm wrong. Also, Singer is a eugenicist dick, but let's not open that can of worms)