They dropped that bit about the DeLorean being extremely cold (after every time travel) almost immediately after that first scene, didn't they?
Not really. It was just not as severe.
When Marty arrives in 1955 we don't get a real good look at the DeLorean until after it crashes into the barn. At which point we can see the moisture and some frost on the DeLorean's surfaces. It's got the dry-ice collection on when it arrives back in 1985 towards the end of the first movie and a little bit of "dry ice smoke" coming off of it when Doc arrives from 2015.
We never see the car well from the outside when it arrives in either 2015 or back in 1985-A but when Doc lands behind the billboard in 1955 the "dry ice smoke" is there. IIRC there's
some "dry ice smoke" when Marty arrives in 1885 and some when it comes to a rest on the rails in 1985.
So only after Temporal Experiment #1 does it arrive covered in the ice and the exhaust vents disperse the condensed gas. All other times the "idea" of the effect is there but not severe. This was because it was costly and time consuming to cover the car in the dry-ice (actually I believe they used CO2 fire extinguishers) after ever time-travel so they mostly just gave the impression with a little bit of dampness on the car to suggest melting ice and some fog/smoke effects.
Here's something to consider:
When Doc drops Marty off at his home in 1985 at the end of the first movie and leaves for "30 years in the future" how long is Doc gone? In theory Doc could've been gone for months or years before coming back to 1985. Given what we find out int he second movie we know Doc had time to do the following:
Collect money from various time periods (as seen in his suitcase, this may suggest time-travel to
multiple eras!), get the hover-conversion done to the car (likely to be a few days work) get the fusion generator installed (also likely to be a few days work) and one wonders how he got both of these things done without the workers wondering about all of the crazy equipment in and on the car including the fission generator. Doc had time to get the "all natural overhaul" of his body including replacement of organs and a face-lift and he had time to thoroughly research the fates of Marty Jr. and Marlene to work out his plan to "save" them.
Doc laments before leaving 2015 that he regrets having never visited the Old West but considering he has 19th century money in the suitcase it seems possible he may have visited there and was just throwing Marty off his scent or something. (And, yes, it's possible Doc acquired the money by other means but I find it hard to believe Doc could easily acquire vintage dollar bills easily unless while in 2015 he discovered eBay.

)
Looking at the BD copy of Back to the Future Part 2 doc has money from the following times:
1955
1923
1917
1934
1902
1864 (two slots)
1985
1914 (three slots for it, in fact)
1863
1861
1866
1875
Each slot has a decent amount of clean, newish-looking, cash in it. So either Doc got some really good-looking vintage cash or his been to all of these time periods. Incidentally, the money he gives Marty to buy clothes with comes out of the 1934 slot.