I've noticed that my Osram NAV-TS 70W Super 4Y, don't warming up faster on my Philips PV C 70 /S CDM electronic MH ballast, than the Osram NAV-T 100W Super 4Y on their magnetic ballast at the Gaash Kochav lanterns, and it seems that the elevated current of my electronic ballast do nothing to the lamp (It still takes 2 mins to reach 80% brightness).
Are HPS lamps requires more current during run-up to warm fast like MV and MH lamps do?

It is not "required" for warm up, but is required for stability - to stop the warm up from continuing when the lamp already is at its intended power

On a magnetic ballast (here i mean a 230V choke ballast, but HX will also have similar behavior), this is naturally achieved by the way how the magnetic ballast determines the current :

The current depends directly on the voltage drop across the inductive impedance. For a choke this simply means the choke's impedance, for HX this means the impedance of the secondary

The voltage drop on the impedance is a vector subtraction of the lamp arc voltage from the Voc. The voltage on the ballast and the lamp are approximately 90deg apart. (It is not exact because of additional less significant factors, like the discharge power factor and ballast wire resistance)

Simplifying things, the voltage drop on the ballast impedance will be lower when the lamp arc voltage is higher, and the effect will be more significant the higher the lamp arc voltage is (due to the angle between the vectors)

When the HPS lamp is fully warmed up the discharge voltage is higher, so the voltage drop on the ballast is lower, so the current is lower. (Compared to the current when the lamp is warming up)



The lamp and magnetic ballast are designed so, that the lamp's full power rating is near the max power which the ballast can provide. If the lamp arc voltage rises more, the current must be dropped significantly, in order for the overall effect to be power reduction and not power rise. This behavior will stabilize the lamp to the correct current/voltage/power and prevent runaway

During the initial warm up it is less important how exacty the current behaves (within some reasonable limits), as it will start and warm up anyway. The magnetic ballast's behavior is fairly simplistic, and cannot be magically divided into ranges of different behavior for lamp warm up vs. full power. So the same relation between lamp voltage and current which is essential for stable lamp operation, continues also into the warm up area

And this behavior is good, as the higher current during warm up accelerates the warm up



With electronic ballast, there is complete freedom to design the ballast to behave in any wanted way, including :
 - Imitating a magnetic choke/HX ballast
 - Current source (imitating CWA)
 - Power source (imagine what a REAL CWA would be which its name implies, but magnetic CWA isn't)
And many more options

But, unlike magnetic ballasts, with electronic it is possible to detect the lamp warm up stage, and provide piecewise combinations of the different behaviors for different stages of the lamp warm up

This way, an electronic ballast may accelerate the warm up, reduce electrode wear, etc. And, independently from this, provide the correct behavior to keep the lamp stable when it is running at its full power

The HPS vs MV vs MH differ significantly in the voltage drop when cold, that implies different real power dissipated in the lamp at that phase.
And each ballast behaves differently, as Ash wrote (with the magnetics the core saturation makes the current even more dependent on the ballast drop so the arc voltage).
So some ballast may boost the current more at really low arc voltages, so lamps operating at lower arc voltages warm up faster than on other
But for lamps with higher cold state arc voltage the difference could be way smaller. Or some other difference in ballast characteristic may even "reverse" the difference (on one ballast lampA may warm up faster than lampB, with different ballast the B may be actually faster than A). Even seemingly "the same" series chokes may behave differently, because with each of them the exact shaping determining the saturation behavior may be different.
And also with magnetic ballasts, difference in mains voltage may shuffle the relation as well (affecting how "deep" the saturation goes)...

Note that the magnetics is usually designed so it does not saturate everything at once (leading to sudden drop in inductance), but very gradually (first saturate just the end tips of the plates which are the closest to the "opposite side", reducing the inductance only a bit, with the increasing current more and more of the plates saturate, making the overall characteristic rather soft). And in shaping of this behavior (carefuly designing the exact shape of the plate ends/tips where the gap is) lies part of the "trickery" to e.g. shape the current vs arc voltage dependency or the robustness against mains voltage variation. And where a lot of difference among various ballast models may be (one optimized for better lamp performance but not tolerating that much mains variation, other designed to tolerate wider mains range but having some compromise in lamp characteristics).