Short Answer
As a practical matter, the strategy you suggest is widely used, and the vast majority of the time, it does not result in copyright disputes. It isn't 100% risk free, however.
Long Answer
All of the questions in this answer are governed entirely by federal law. Both basic copyright law, found in Title 17 of the United States Code, and the Digital Millennium Copyright Act (which amended Title 17 of the United States Code in part, and also added 28 U.S.C. § 4001 related to movie licensing) are federal statutes that do not have parallel state law counterparts.
Copyright cases are in the exclusive subject matter jurisdiction of the federal courts when they arise in the United States. See 28 U.S.C. § 1338(a) ("[t]he district courts shall have original jurisdiction of any civil action arising under any Act of Congress relating to ... copyrights .... Such jurisdiction shall be exclusive of the courts of the states in ... copyright cases.").
This isn't a DMCA case
This strategy doesn't really utilize Section 230 of the Digital Millennium Copyright Act (DMCA), however, which protects online hosts of sites from liability for user generated content, if the host honors DMCA takedown notices (subject to an exception for human trafficking content). DMCA takedown notices aren't strictly speaking applicable in this fact pattern.
When Bob is the website's author, his content isn't user generated content from which he is protected from liability by Section 230. His Internet Service Provider is protected from liability for his content, but Bob isn't protected from liability for his content, because he isn't a third-party user of his own website.
The Basic Copyright Law Analysis
Once you understand that nuance that this isn't a DMCA question, the question collapses into a simple copyright infringement issue.
The bottom line in this copyright analysis is that if you take a snipper of material from a copyrighted source and link to the source copyrighted material, are you not likely to face copyright liability.
Links Are Basically Liability Risk Free
The link itself, pretty much categorically, and without lots of additional relevant facts (basically some sort of conspiracy behind the scenes that can't be discerned from looking at the work itself), will never give rise to copyright infringement liability.
Snippets Are Copyright Infringements That Fair Use Usually Excuses
So, the remaining issue is whether a snippet of a copyrighted work flagged for educational purposes, creates risk of copyright infringement liability.
There is always some risk any time you publish a copy, no matter how small, of copyrighted material without the permission of the copyright owner.
The snippet is undeniably a publication of a copy of copyrighted work (by virtue of the premise of the question). This is a prima facie case of copyright infringement and your inclusion of a link to the source makes it fairly likely that your publication of this copy will be discovered by the copyright owner. So, this is a copyright infringement that gives rise to copyright liability unless this liability for copyright infringement can be overcome or excused by the affirmative defense to copyright liability of fair use.
The risk that the snippet will not be considered fair use, and therefore will be found to be a copyright infringement giving rise to copyright liability is relatively higher if the snippet is a photograph or a sound recording than it is if it is merely quoted words from the source.
But, usually, a reasonably small snippet of a copyrighted work published for educational purposes will come within the fair use defense, even though that really needs to be evaluated on a case by case basis.
Practical Context And Analysis
The fact that this mode of expression, with a snipped and a link on a blog for educational purposes is pervasive on the Internet, as a practical matter, means that generally speaking, this will be considered fair use.
Usually, this strategy will prevent someone who uses it from facing litigation for copyright infringement, even though, in theory, the determination is highly fact and context specific.
It isn't a guarantee of no liability, but it is low risk.
Most marginal cases where someone arguably slightly exceeds fair use are resolved with a cease and desist letter (with which Bob should comply even if he thinks his use is fair use), rather than litigation.
