It's definitely not IPS returning the error itself. That is not their style of error message, etc.
The question is what is actually returning the error... for example, can others replicate that same issue? It's possible there is some sort of home firewall or ISP home protection service that could be blocking the request.
There could be a firewall at the ISP side. You mentioned site ground as your host, but when from the video you posted, it looks like the domain in the video is pianolessonwithwarren.com which according to a WHOIS lookup is using Google's Cloud Computing infrastructure. (It's possible Siteground is using Google for hosting similar to how IPS uses AWS for it's cloud infrastructure.)
I use Cloudflare's WAF to protect my own website... and there are occasions where I have to optimize a WAF rule because it unintentionally blocks replies with large amounts of text/links, again thinking it's something malicious. Firewall rules are super sensitive about content linked from 3rd party services. Since you mention it happens only with quoted videos, that makes sense. Lots of HTML styling is going on in the background around the video and the video itself is most likely on a hosted service. So all of that together could trigger a WAF to think something was wrong.
Ultimately the question is where is this WAF in between you and your site... is it on your computer, at your ISP, at your hosting company, etc. I would ask Siteground if they use some sort of Web Application Firewall they use with their hosting service. In Google searching Siteground WAF, I see a couple of results that would lead me to suspect they might have something there...
https://www.siteground.com/blog/new-and-improved-sitescanner-security-service/
You should also try seeing if others see the same behavior as you do... if so, it's something at the server itself. Otherwise it's something either on your computer or on your home network blocking it.