The key difference between a TVL E13 and E14 position would be the job responsibilities and required qualifications. In an academic context, E13 positions are for researchers at graduate or post-graduate level working under supervision. E14 positions have a higher responsibility in the sense that these would typically include project leadership and / or supervision tasks. Typical academic roles that should be filled at the level of an E14 position would be group leaders, principal investigators of projects, research lab managers, or just someone supervising graduate or other post-graduate researchers. Nevertheless, someone working in these roles may still get only E13 due to budget reasons.
If the position you have an offer for includes tasks at E14 level, then it certainly is something that can be asked for. Whether or not it will be possible to make it E14 will mostly depend on how the position is being funded. It may be that this was specifically budgeted as an E13 position, and then it will be difficult to change that through negotiations from your side. If the position was announced specifically as E13 then it may be a high administrative hurdle to change that. However, if the funding is just a bunch of money without detailed budget plan, it may be easier to get it as an E14 position. Especially if it is only for few years, the total salary spent need not be much higher in E14 compared to E13 because you would likely start in a lower grade (Stufe).
Regarding the grade I think that indeed with more than 8 years of research experience you should definitely get Stufe 3 in E13. The number of years could even be enough for Stufe 4, but I've heard that new hires will usually get Stufe 3 at most. However, if you get an E14 position, then I guess that you can just get Stufe 1 or maybe 2, depending on whether you have some prior experience with the E14 type of tasks I mentioned above.
As a side note, there is also the possibility to start in E13 and then being promoted to E14 if job responsibilities are accordingly and the budget is available. In that case, one would get the lowest grade in E14 where the salary is still at least as high as in the most recent E13 grade one has achieved.
