Did Germans and Soviets count casualties differently at the Battle of Kursk?

The 863K losses come from Krivosheyev's research. These are very precise numbers and unless one is a professional historian who did research in this specific field, he cannot objectively question these numbers. The research Krivosheyev did is monumental and acknowledeged worldwide (see Glantz). That precision is absolutely sufficient for discussions online.

Kursk defensive operation, 5.07.-23.07.1943

Divisions — 77, corps — 9, brigades — 14, Fortified Regions-3;  Total manpower: 1.272.700.
Losses: irrecoverable - 70.330 (5,5 %), sanitary - 107.517, total - 177.847,

Operation «Kutuzov», 12.07.-18.08.1943

Divisions — 82, corps — 8, brigades —14, Fortified Regions-3; Total manpower: 1287600
Losses: irrecoverable - 112.529 (8,7 %),    sanitary - 317.361, total - 429.890

Operation «Rumyantsev», 3.08.-23.08.1943

Divisions — 50, corps — 11, brigades — 5; Total manpower: 1144000.
Losses: irrecoverable - 71.611 (6,20 %),    sanitary - 183.955, total - 255.566.

Total, 5.07.-23.08.1943

Divisions — 132, corps — 19, brigades — 19, Fortified Regions-3; Total manpower: 2431600.
Losses: irrecoverable - 254.470,    sanitary - 608.833, total - 863.303

So 863.303 losses are total losses of all causes for all the armies involved in battle throughout the period, considered by the Russian historiography. Since Western historiography considers a smaller period of the operation, many, often willingly, try to compare German irrecoverable losses (this question is the case BTW) to Soviet total losses within different time slices. German losses often lack their allies, various collaborationists etc. Overmans claims that for July and August 1943 Germans have lost 130.429 men killed. If "killed" means it and is not "irrecoverable" loss, then this number should be even greater, since there are more wounded than killed usually. Higher Soviet casualties are explained, roughly speaking, by the following factors:

• general superiority of Germans in artillery: they had more high caliber guns, more barrage mass and were usually winning artillery duels. At Kursk, North part, the Soviets were expecting a stronger blow, so they amassed huge amounts of artillery there. This was the main reason the offensive went much harder for the Germans there. This however was not the case with the Southern part. Artillery is the most notable factor in losses.
• worse Soviet tanks and organization of tank forces at this stage. Up to half of tanks were T-60 and T-70. T-34s had an insanely tough time fighting German heavy tanks, should they encounter those. Elite German tank units were indeed better trained. However they were irreplaceable in all-in war realities.
• air superiority problems for the Soviets.

First:

Is propaganda on either side a strong force for these numbers?

Yes, the table of causalties for Kursk is strictly debated because there was a lot of propaganda involved.

On the Soviet side: Contrary to Stalingrad for example, where a Soviet dead soldier could be given as a "victim" of Nazi's cruelty, defending civilians in a besieged city, the battle of Kursk is supposed to be a "military-only" fight on an open ground: some sort of duelling between two armies, with fair conditions (no winter is an important factor for Russian propaganda: winning Kursk means the Red Army won the war, and not general Winter).

On the German side: Same idea of duelling without winter, and you could add another factor: It could be seriously argued that Kursk was the last chance for Germany to win the war, so Soviets having very-very high losses could temper the fact that, at the end, Kursk is a Soviet victory because Axis attacks were repelled.

Second:

Do Germans and Soviets count casualties, particularly for their armor and wounded, very differently?

Yes. The count is different for armor because of multiple factors:

• Kursk, for the more restrictive part (German point of view), is only a matter of Axis attacks. During those attacks, German captured the ground on which they fought. So they always managed to get back the tanks and vehicles they had lost for minor damage, such as driving on a minefield. Those tanks were counted as damaged and not destroyed, so they do not enter the Casualties Table.
• Moreover, German repair teams were numerous and German heavy tanks were so armored that they could not be destroyed with one hit: they were always reparable
• However, those damaged vehicles needed time to be repaired: sometimes so much time that they did not enter the battle again before Hitler called off Operation "Zitadelle".

All of those factors are not applicable for Soviet armored vehicles.

In the end: Germans had few destroyed vehicles, but a lot of damaged ones. So much that came a time when they had no more forces to go on the attack, and the Soviets were counter-attacking. A lot of movements for German panzerdivision and their repair teams, associated with lost ground, led some difficult but reparable tanks to be destroyed.

Theses factors explain why very loopsided casulaty ratio in favor of the Germans is not really a lie, but is neither representative of the final situation.