I've seen a few types of consultants.
One type, to borrow from Ken Blanchard, are "seagull consultants" who fly in, make a lot of noise, dump on everyone, then fly out. You'll often find them flocking around ERP and CRM implementations. They come in, insult the staff and leave without really doing much.
Another is the body shop consultant. This usually happens when a company wants to put the "Mythical Man Month" to the test by adding more developers to a project that's already behind. Here you'll find a lot of underpaid H1B workers and those willing (aka desperate enough) to take a low paying temporary gig with toxic management.
Both of those are the usual result when management has become frustrated with their in-house team. But there are positive ones...
The utility consultant is someone brought in to fill a role temporarily. For example, a legacy application might need support and maintenance while a new application is developed. It's not a bad gig if you don't mind working with older tech (like VB6 or Powerbuilder) and if the original system isn't a horrific mess.
Project consultants have a similar role. They're used to complete a specific project that the current in-house team doesn't have but the company doesn't want to hire someone perm to do for one reason or another. For example, a company that has an iOS app might bring in an Android consulting team to do a conversion.
Expert consultants are typically engaged when a business needs someone to coach their team on a particular area they need help with to move forward. For example, a team moving from desktop to web based applications might need someone to help them get over the initial learning curve.
A good list, but I'd add one more:
The Scapegoat/Punching-bag consultant is someone who can come into a team locked in a stand-off between management and the rank-and-file to make the "unpopular" decisions. This can be anything from recommending a large-scale reorg, killing-off a project, or (in the worst case) layoffs. By bringing in a consultant to do these things, management gets to save face while the rank-and-file blame the "damn consultant".
I briefly worked as a "strategy consultant" it's amazing how much people will pay not to deliver the bad news
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