Trying to pick a neighborhood

Maybe check out La Moraleja, just outside Madrid. Lots of expats live there because of international/bilingual schools. Lots of gardens, semi-gated communities, chalets. Not cheap though and honestly a bit posh (like the Salamanca neighbourhood in the city centre).

If you like trendy neighbourhoods Malasaña and Chueca should be on your list.

But honestly if you like quiet you should avoid the city centre all together because the Spanish come alive at night and they have a habit of talking quite loudly  .

Good luck with the search!

The OP didn't mention kids / school so  not assuming anything.

Before settling down I stayed in different areas for 4-5days each and got a feel of everywhere. From the age range and your preference for close to the city but not in the thick of it, Chamberi would be the top choice. In fact I also liked Trafalgar the best when I saw the big 24/7 Carrefour! I took a look of Salamanca but I found the residents too old for me. Malasaña and Chueca are too touristy and young.

Recoletos is beautiful and green and next to Parque de Ritero as you know but it's too quiet for me personally. More restaurants bars and supermarkets in Chamberi..Streets in Palacio are really beautiful with old architectures and historical streets but the layouts of the old buildings are just not practical with many windowless rooms. Embejadores is the counterpart of Chambri on the south side of Centre but just a tiny tad lower scale. Still pleasant and convenient and very accessible to the Riverside park along Madrid Rio.

Chamberí is a great neighborhood. I would also recommend Salamanca. It's a quiet neighborhood close to Retiro, one of the most beautiful parks in the city