Preamps built Into DACs

In English language bunch of different words, expressions may mean the same thing and yet the same word can have a tons of different meanings. These factors you should know in order to stay within our 3d without going beyond.

DAC with volume control

Preamp with digital input(s)

Full function preamp(usually applied to the ones with phono)

DAC-preamp

preamp-DAC

...will basically mean the same and share the same idea

Conserning your second paragraph, I believe you're tripping or indeed found yourself in multi-d space and have no idea how to get outahere. Saving money by removing DAC? or preamp? How about removing DAC from CD-player? I don't getitatall. I've always believed that one box will be cheaper that two boxes.
One preamp with DAC and phono will keep you from necessity buying both of separates...

I used to study the multi-d math, but kinda quit doin' that coz got tired understanding those theorems that are necessary for you not to get lost.
To continue studying multi-dimentional calculus I would probably have to smoke an ounce of MaryJohana per week.

Mitch - Beware replacing a good active pre with a volume control in a DAC. Most of them are substandard and will be a step down.

On the other hand, a few are excellent and will probably beat your active pre. Depends on the technology and implementation. It varies widely. I would read the reviews. If they say it beats good preamps, then its probably a good candidate.

Some are based on variable gain control of an active buffer stage. Others are based on resistive attenuation combined with a gain stage. Some are based on using an integrated chip to digitally control the gain of an analog stage. Two of them are based on controlling the reference voltage to the D/A conversion.

The last 2 generally deliver the best SQ, but again it depends on the implementation, the design details. Controlling the D/A reference voltage is the only one that actually reduces distortion as the volume is decreased.

You can also use purely digital volume control, but I dont recommend it. More than about 9 dB of attenuation will start to reduce sound quality.

Another option is to use a passive transformer based volume control (TVC). You basically cannot lose doing it this way and it beats pretty much all active preamps.

Steve N.
Empirical Audio

This is a good question, the ps audio perfectwave dac mk2 has a preamp section in it, wondering how good it could be

Grace M903, Mytek 192 DAC-preamp, Antelope Zodiac+, Lynx HiLo modern professional DAC-preamps have superior preamp section within the price of $2000.

In general, my only concern about a DAC and pre-amp in a single box would be how well is the pre-amp section shielded from RFI generated by the digital circuits/DAC? In general, I would prefer these in separate boxes separated by space as an insurance policy at a minimum. I would definitley not try to cram a low level phono section in there as well.....

A similar concern with integrated amps is proximity of power amp with transformers, etc. to the pre-amp section, again especially with phono. HEre the issue is more EMI than RFI perhaps, but I would prefer separates here as well to physically separate the two as an insurance policy at minimum. Often, noise levels in circuits can affect the sound quality in subtle ways well before any effects are clearly audible.