One thing I should say is that while Martinez/San Isidro is a little less dense than some areas of Capital it's not exactly suburbia. The houses may be larger but the properties are small so if you're really after outdoor green space with generous lawns you need to go to Nordelta and the barrio cerrados -- most of those places started precisely as weekend getaways so the lot size is much more generous. You'll absolutely need two cars if you live in one of the barrio cerrados, and I'd even say you'd probably want two if you were in the Zona Norte. In the barrio cerrados while you may think the surroundings are pretty your spouse will go insane being trapped there with basically nowhere to go, a car is an absolute must and both of you will just have to get used to crazy traffic if wherever you're coming from doesn't have it. I just think ugh, with a couple of kids doing activities you end up spending half your day chaufffering people around.
La Lucila is most likely the other place that LaCoqueta was thinking of. It's an area with very nice houses but again, let me explain, it's not suburbia with enormous green lawns. Your neighbour's house is going to be backing yours pretty closely. I like the Acasuso area as well.
But look, again I underline, there's not really somewhere out there with an international bubble of expat families sending to Lincoln. Contact the school. I'll be surprised if 15% of the student body is foreigners. And if most of those are here for 18-24 month contracts what does it matter, these aren't really going to be friends. Argentines can be difficult to get to know but it is true, at least if your kids get Argentine friends they will be more accepted and those are friendships that are likely to be deeper than with international kids who are just biding their time until they get to go home.
From what I've seen from people working in the embassies they tend to get placed in apartments in Recoleta / Palermo areas. If they have kids a lot go for Belgrano. And then yes there are a few that go for the northern suburbs. Contact the school. They are your best resource for finding out where to live. And if they don't recommend anything contact your embassy.