DC bias and vertical offset are the same thing. Only the representation on the display is different. You won't find two different signals to set the DC offset / vertical offset when you are going to look for them in the analog frontend.
They might be two different signals. It will depend on the design.
Offset and vertical position do the same thing, and the later may be used as the former like Tektronix likes to do, (1) but offset is added before some or all of the attenuation, and the vertical position is effectively added later, (2) or implemented digitally. This gives the offset more range than the vertical position, and a range which depends on how much vertical attenuation is in place after the point where offset is added. In extreme cases, the offset control can have a position range of 10s of thousands of divisions.
(1) On older DSOs Tektronix made the displayed signal only part of the digitizer range for greater overload capability and to prevent vertical aliasing in the display. Typically 25 counts per vertical division are used, so 8 vertical divisions represent 200 counts and there are 55 counts not shown. This prevents a minor overload from affecting anything, and signal processing can correctly process a further division above and below what is shown on the display. Vertical aliasing is prevented because there is a direct correspondence between the digitizer counts and screen pixels. I do not know what Tektronix does on their current DSOs.
(2) I have seen some designs where the vertical position was added early in the signal chain with variable attenuation to make up for later vertical signal attenuation, and of course some use a real offset control for position, instead of the reverse like Tektronix.