I used to work for a big audit international audit firm. A "senior" is a person who is in charge of a team of "juniors" and is hoping to become a "manager" one of these days. A "manager" is someone in charge of a team of "seniors" (and all their "juniors") and is hoping to become a partner some day.
They were generally referred to as "senior" but in written proposals they would be called "jefes de equipo". The "juniors" were called "consultores" (!), at least for written proposal purposes.
The whole "engagement" thing means "el proyecto en cuestión". It is used to disguise the fact that most of the people assigned to a project (e.g. audit, due diligence, etc.) are also working on something else too.
So, you can't call a senior "gerente" because that's what his/her boss is called.