The short answer
Mostly, yes. Staff augmentation, team augmentation and resource augmentation are used interchangeably for bringing external engineers into your team under your direction. Where a vendor does distinguish them, team augmentation tends to mean adding a group of people at once. In every case you keep ownership of the work.
Mostly, yes. All three terms describe the same arrangement: external engineers join your team, work under your direction and are billed for their time, while your organisation keeps ownership of the work and its outcome.
The wording varies by vendor more than by meaning. Where a provider does draw a distinction, team augmentation tends to mean adding a group of people at once rather than individuals one at a time.
The distinction that actually matters is a different one: who owns the outcome. If you direct the work, it is augmentation under any of these names. If the provider owns delivery of a defined outcome, it is a managed service or a delivery engagement — a different contract, with different responsibilities on each side.