Correct; "Only one grenade (of any type) can be thrown by a unit per phase."
That specifically refers to the shooting phase.
We know this because two paragraphs further on that exact same page (180 of the hardcopy, or 3rd paragraph of the grenades section otherwise) it states;
"A model can use such a grenade as a Melee weapon, but can only ever make one attack, regardless of the number of Attacks on its profile or any bonuses."
So shooting is 1 grenade/unit.
And assault is 1 grenade/model.
Hope this helps!
I dedicate these deaths to Odin Allfather, Spearshaker, One Eye.
Fleet Shadowmaker - Kabal of the Dying Sun; Cult of Marrow Excised; Coven of Lambent Hunger
Sons of the Last Breath - Chaos Space Marines
Host of Shattered Purity - Chaos Daemons