I dunno about today record/DJ/converter players and their stylus'....but back in the good old days the stylus' were graded as soft, medium, and hard......I think from memory sapphire stylus' were medium/general purpose while diamond tips were hard for higher fidelity and crisper high frequencies (although at the risk of higher record wear as the diamond tends to cut your record after many many plays).

Old records on an old record player will always have a warmer/richer sound than CD's as there's 2 VERY different ways the audio is being processed.
It must be noted that records sound 'warmer' due to the lack of very high frequencies, these frequencies are cut out during the record-cutting process, where a filter (aka RIAA curve) rips out a band of high-end energy as part of the surface noise reduction (eliminates the physical noise of the styless scraping along the vinyl).
The RIAA filter also cuts a bunch of bottom end bass out, as too much bass signal will cause the left/right tracks to crossover into eachother, so an instrument on left channel would also cut into the right channel if it's too bassy/loud....
The filtering of bass maintains that each of the 2 cutters will ONLY cut information into their respective left/right track areas in the record groove.

Just about every (old) amplifier/phono-input has an RIAA preamp which does it's best to boost back the bass & high frequency response along with using the Graphic EQ to also adjust the sound to your needs.

CD encoding is a different can'o'worms, a lot of information HAS TO be lost during the analog to digital conversion to write the CD audio, as there's too much high frequency energy for the encoders to handle, so they cut it out to reduce bit-rate errors during playback.
This removal of high & low frequencies isn't 'compression, it's actually 'deletion', but most of the deleted info isn't exactly audibly noticable, but it makes a BIG difference to the data being send to the digital encoders without dropouts..

CD inputs on amps have a wide dynamic range and also re-instate/boost the highs & lows to TRY to bring the CD audio back to full quality, but with all the pre-filtering during encoding, CD's just won't match a record.

Also to note, old records usually being played thru old amp/hifi systems that have preamps catering to analog signals and RIAA factors, you would find that plugging a CD player into those old amps would actually sound a little warmer too.

I recently sold an old radio/stereo record player (radiogram) which had valve amps in it, i played a record for the buyer to prove it worked etc etc, and as soon as i heard the thick warm tones i started to think about keeping it, *if* the buyer changed his mind.....
Sadly he bought it, but i'd since gained some old valve radios & amps to fill the gap.

MM