I'm a Dummy, Tell Me About Turntable Mats

@lewm, it is not an echo. It sounds like a very high frequency copy of the music. What is vibrating can not emit anything from maybe 6 kHz down. I'm just guessing. With the vacuum engaged I can just barely make out cymbals. With the record loose it is 2X louder and I can make out a little more. It is still very quiet. You have to put your ear next the rim of the record to hear it. I remember my old Zenith which I got when I was 4 years old. I use to marvel how I could hear the music with the volume turned down. It was pretty loud. You could hear it across the room. 
Remember the old Victrolas? They had a sowing needle in a shank fixed by a grub screw. The shank was attached to a diaphragm in the throat of a horn. Same thing but no diaphragm or horn. You are hearing whatever is vibrating. The less you hear the better is the control over distributed resonance from the stylus and cantilever vibrating. The cavity around the stylus can act as a horn which is why many manufacturers stick the cantilever out in thin air. That resonance is reflected back at the cantilever to be heard. You really only want to listen to the record. Nothing else.

The best record playing set ups hardly make any noise at all. There may be a few out there that make no noise at all. I have never heard one make absolutely no noise but I have heard a few get very close. 
Just don't laugh when you see a guy with a tonearm stuck in his ear. 

The point of all this is, the job of a turntable mat is to help control resonance. It is to help keep the record from vibrating under the stylus.
Because records are not flat a mat can not do it alone. People figured that out and started using record weights, then record clamps, then taking a lesson from record lathes, vacuum clamping. I am comfortable with saying that vacuum clamping is overall the best. I do not know if it sounds better than reflex clamping but it does results in the best control over the record with the least effort and stress on the record. 

If your definition of record noise is “a high frequency copy of the music” then we’re probably both talking about the same phenomenon. I find it varies in audibility in relation to cartridge output voltage magnitude. I have always felt it comes with the territory, and I pay no attention to it, because my listening seat is far enough away from the turntable that I never hear it while I am trying to concentrate on what is coming out of the speakers. What’s the big deal?

No felt. No tape. No weight. No problem.

@lewm, It is not really record noise. You are hearing the mechanical system of the record/stylus/cantilever/tonearm vibrating. Ideally, all the energy spent in moving the stylus should be translated into an electrical signal. Some is turned into heat and some is turned into noise. The noise is a problem because it reflects back on the structures creating it which you theoretically can hear amplified along with the music. 
Let us call this "tracking noise." Tracking noise has been mentioned by others as a measure of the quality of the playback system. Certainly, those old ceramic needles were very loud and I remember my fathers old Rec O Cut with its old Empire cartridge was easily audible 3-4 feet away.
My current combination is very quiet but my main point is that clamping the record properly makes an obvious improvement. 
I had a Transcriptors turntable at one point decades ago, the one that set the record on 6 brass weights. I hated it. It was pitiful in comparison to an LP12. I blamed the platter for that.
I'm sure Chakster and Rauliruegas can comment on this.

Mijo, You wrote, "It is not really record noise. You are hearing the mechanical system of the record/stylus/cantilever/tonearm vibrating."  How does that contradict anything I wrote?  Moreover, your statement itself is a bit contradictory internally (to say it is not "record noise", which was your term, not mine, and then define it as in part due to mechanical energy due to the record, along with the other factors you name).  Anyway, I view it as a "problem" endemic to vinyl reproduction, and it probably varies all over the place based on the tonearm design and construction and the cartridge design and construction, and I have better things to worry about. Is there any test or are there any measurements that correlate SQ with reduction of this "noise"?  In other words, if you believe it to be undesireable, what does one do to reduce it, and does that result in better SQ?