First of all, the relevant rule here is § 16(2) Bund or a similar rule of a similar tariff contract (translation mine):
¹Bei Einstellung werden die Beschäftigten der Stufe 1 zugeordnet, sofern keine einschlägige Berufserfahrung vorliegt. ²Verfügt die/der Beschäftigte über eine einschlägige Berufserfahrung von mindestens einem Jahr, erfolgt die Einstellung in die Stufe 2; verfügt sie/er über eine einschlägige Berufserfahrung von mindestens drei Jahren, erfolgt bei Einstellung in der Regel eine Zuordnung zur Stufe 3. ³Unabhängig davon kann der Arbeitgeber bei Neueinstellungen zur Deckung des Personalbedarfs Zeiten einer vorherigen beruflichen Tätigkeit ganz oder teilweise für die Stufenzuordnung berücksichtigen, wenn diese Tätigkeit für die vorgesehene Tätigkeit förderlich ist.
¹On hiring, an employee is assigned Level 1 unless they have relevant work experience. ²If the employee has a relevant work experience of at least a year, they are hired at Level 2; if they have a relevant work experience of at least three years, they are assigned Level 3. ³Irrespective of this, for purposes of finding personnel, the employer may completely or partially recognise times of previous work for assigning levels, if this work is beneficial for the designated employment.
So up to Level 3, there are no constraints on where you obtained your work experience.
(Note however, that experience within Germany’s public service can get you hired at levels higher than three as per §16(2), 3 or § 16(3).)
However, there is a large wiggle room as to what counts as a relevant work experience. This is where Lighthouse Keeper’s answer comes in: You have to convince some bureaucrats what your previous work experience is worth. In particular, I am not surprised that they won’t count industry experience.
That being said, your situation is somewhat special: The vast majority of positions from PhD positions to staff scientists are Group 13. However, you got one of those very rare Group 14 positions. That means that you get more salary than a regular postdoc (Group 13) with the same experience. In particular, starting at Group 14, Level 2, your accumulated salary will always be higher than if you had started at Group 13, Level 3 (see below why this is a relevant comparison). On the other hand, this is a good argument for not counting typical PhD and postdoc positions, since those are not a Group 14 experience. On yet another hand, you say you have team-management experience, and team management is exactly what usually elevates Group 14 from Group 13.
Therefore I expect that arguing for Level 3 in your situation will be difficult, though not impossible.
Sidenote: Why some experience does count for Group 14
You may wonder why you got any experience accounted for at all (starting at Level 2 instead of Level 1) and you also report:
I understood that their experience as a PhD student on E13 counted in E14 as a postdoc
The reason for this is that being hired into a higher group cannot be disadvantageous for you. If you had been hired in Group 13, you would very likely have started at Level 3 (the highest non-exceptional entry level as per the above). The salary of Group 14 Level 1 would be lower than that, and thus you were hired at Level 2.
The same likely applies to those PhD students you know: If they previously were employed at Group 13 Level 3 (on account of their experience during their PhD studies), they could not be hired at Group 14 Level 1, because it would have been detrimental. Thus their experience as a PhD student did count.
However, to achieve Group 14 Level 3 by the same mechanism, they would have to have reached Group 13 Level 4 as a PhD students, i.e., they would have to have worked on their PhD for six years. That’s already quite some experience and also only possible to obtain within the German system or by invoking the exception from Sentence 3 of the above law.