for turf grasses, i think a botanist might be able to get you to a species (ex. Cynodon dactylon), but the landscaper who picked the seed or sod rolls / squares should be able to get you to a variety (ex. Tifway 419 Bermuda). that said, in most lawns, if i look closely, i’ll see lots of other species that have popped up within that main species, and sometimes, the landscapers will even purposely seed, say, a winter grass that will look good when the other grass has gone dormant.
since i’m not a grass expert, i have to rely on keys to help me identify. just looking at some of the commonly observed Poales in Israel (https://www.inaturalist.org/observations?place_id=6815&subview=table&taxon_id=47162&view=species), it looks like a lot of these are common even in my area (eastern Texas, USA).
the resource that i usually look at first for grasses in my area is Volume 1 of the Illustrated Flora of East Texas (http://brit.org/brit-press/easttexasflora) or the slightly older Illustrated Flora of North and Central Texas (http://brit.org/brit-press/nctexasflora). in particular, these contain a Subfamilial Phylogeny/Classification of E TX Poaceae (appendix 4 in http://brit.org/sites/default/files/public/BRIT%20Press/IFNCT_Docs/SBM_26-pp1153-1509.pdf), as well as illustrated keys (http://brit.org/sites/default/files/public/BRIT%20Press/IFNCT_Docs/SBM_26-pp309-1151.pdf).
if the keys noted above are not sufficiently detailed, i usually then turn to efloras.org, which has keys from Flora of North America, as well as Flora of China and others. the FNA keys don’t usually have illustrations, but the keys, if they exist, are generally very, very detailed. the FoC keys are sometimes illustrated and usually have fairly good detail, too.
that said, the keys noted above aren’t specific to your area, and i couldn’t find anything similar for your area with a cursory English-language search online. but i did find a Centre for Middle Eastern Plants (http://www.cmep.org.uk/) that maybe you could reach out to for more information, or maybe some of the faculty at your university or other local universities might also have some good ideas for resources.