1st, I think we need to assume that neither German High Command nor Spanish government was so stupid that they had difficulty telling which unit belonged to themselves and which unit not.
Well, then as the German High Command did not consider the unit as German and the Spanish government did not consider it Spanish... it seems that the answer to your original question is: neither.
2nd, then one side must have been deliberately covering up the facts they held. Judging by motivations:
a. The Germans mentioned the Blue in their INTERNAL operation plans, not propoganda plans, thus they had no motivations to cheat themselves when making such plans.
Well, "motivations" are sometimes much feeble that you think. For example, it could well be that the German High Command did not consider Spaniards as worth of being part of the German Army. They could have considered the division as a mercenary unit, and as such unworthy of being considered on par with the rest of the German Army.
b. The Germans had trained/equipped/paid for the Blue as an allied unit to fight alongside German military, just like the US military did the same to ISF and ANA.
No, this is just your assumption. The Germans had trained/equipped/paid for the Blue to fight alongside German military. It was clearly not a "German" unit in the strictest sense, but it was part of the German Army, not like those Romanian, Italian, etc allied unit and not like those ISF and ANA that were not part of the U.S. Army.
c. It would impair German control over a German unit if the Germans deliberately labelled it as an allied unit, thus against their best interests.
Actually, labelling the 250th Infantry Division in any way would not impair its control by the Germans in any way. Actually treating it as they treated those allied divisions would, but that was not the case. Their best interest was to have complete control over the division when it was on operations, and that they had regardless of the labelling.
Further, it would again impair German military strength to withdraw a veteran fully equipped and trained German unit from a losing and increasingly outnumbered battlefront at the request of a foreign government, when Germans were so desperately needing soldiers. It would be again going against German interests.
Yes, the Germans needed soldiers, but they also needed raw materials. Then, it would again be going against German interest to enrage one of the few countries that still traded with them in 1943. It was a matter of what was more valuable, a infantry division or a exporting country.
So, the Germans had no motivation to cover up anything.
Neither had Spain. Come on, we are talking about a country that did not even try to maintain their appearance of neutrality.
The Spanish government had several motivations to cover up the truth. First while it wanted to fight Soviet Union,
Hardly. The Spanish government may have despised Communism, but their interest in fighting the Soviet Union was philosophical, not real.
it did not want to fight the Western Allies at the same time,
Actually, at the time the division was formed, Spain's interest was in fighting the Western Allies (Great Britain, that is). That was the only way to achieve all the territorial objectives that they wanted. Of course, there was no way they could do it, but we are talking about desires, not possibilities.
which would be inevitable if Spain openly declared war on Soviets, given the precedent of Finland.
Talking about Finland, it will be interesting to know how the German High Command labelled Finnish units, given that Finland was not an allied country, but a co-belligerent.
Then later in the war when Axis defeat was becoming likely, and after the war, the Spanish government wanted to distant itself from the losing Axis, and thus they had the motivation to deny a Spanish unit was involved in the war against the Allies.
But later in the war, it would have been too late to deny anything. And at the beginning, every detail of the division's activity was heavily publishised, hardly the best way to maintain a facade.
Thus, the Spanish government had strong motivations to cover up the facts.
We are talking about a country that did not maintain a facade of neutrality, but adopted a position of non-belligerence. But anyway, it really doesn't matter.
As for the division's oath-taking, it was just part of the covering up, the same as the German title of 250th Infantry Division given to the division.
Perhaps you have little regard for your oaths, but I can assure you that the Spanish and German military at that time were very aware of what they swore. No way this was a cover up, and in fact it was no need for it. It was made because everybody understood what they were: a unit of the German Army, if not a German unit.
The Spanish government was very keen to keep the facade.
Which should make you wonder why the Germans were kind enough to collaborate with all this play-acting. Seriously, this is Germany in the summer of 1941. They are the victorious overlords of Europe, they have no need for an additional infantry division, much less so one in which they have to invest precious equipment and training time of their own, and regarding allied units they had more than enough. Why would they collaborate to maintian this supposed "facade" for an infantry unit of dubious quality over which they would have no absolute control?