Reputation vendors use 'removal', 'deindexing', and 'suppression' as if they are grades of the same thing. They are not. Each addresses a different problem, and using the wrong one produces expensive silence.
Deindexing
Deindexing removes a specific URL from Google's search index. The page still exists on its host, but Google will no longer return it in results. The paths are narrow: publisher agreement, court order, Google's own removal tools (for exposed personal data, doxxing, non-consensual imagery, and similar categories), or a right-to-be-forgotten petition in eligible jurisdictions.
Suppression
Suppression accepts that a URL will remain indexed and instead crowds it out with authoritative material that ranks above it. It is slower, more expensive, and requires ongoing maintenance — but it applies to the majority of situations, where deindexing is not available.
The rule of thumb
Try removal or deindexing first, and only where a legitimate path exists. Move to suppression when those paths are exhausted or unavailable. Any vendor who sells suppression before completing that analysis is optimizing their invoice, not your outcome.