To be correct they price options off the last sale of the Russell future that best covers the option expiration + base offset. You always price your options/greeks off ATMF (at-the-money-forward) price, it's just how you arrive at this price is different. So future price + base offset is just one of the models. Also there're different flavours of delta like gamma-adjusted delta (useful for delta-hedging), skew adjusted delta (or skew-delta, useful for vol trading). It may be that TOS publishes raw BS delta while OptionVue publishes gamma-adjusted delta.
Part of it is attempting to achieve a certain ratio between delta and theta. It involves closing a position if the ratio becomes undesirable. If TOS's delta values are very far off or inaccurate, it becomes impossible to take the trade off at the right time, thus negatively impacting the trade in a big way. The positions just seem to perform better on OptionsVue using their delta and theta values and perform poorer on TOS. You use optionvue in the past tense. Why do you no longer use OptionsVue?
You can change the vol settings in TOS, which will also change your deltas. Also, there is no "one correct delta". It's only an assumption based on your model and volatility input.
I use both TOS and OptionVue. Yes the deltas are different and that's because TOS and OV model their IV differently. For most people, that difference doesn't really matter but for some who trade volatility based strategies or delta neutral strategies, such differences do matter. A better model is one that offers predicted results closer to reality. In my experience, OV provides closer forecasted values than TOS. I use both in my trading. OV to analyze and TOS to track my position. When I look at deltas on TOS, I'll have to do some mental adjustments to correct. But I'm already very used to the differences. So that's no big deal. However, to be fair to TOS, they are a broker and not a software vendor. In my opinion, TOS platform provides the best option analysis capability among brokers. But their IV modeling is still pretty basic. That's understandable. OV on the other hand is an analysis software. Therefore, they are expected to be better in modeling. There are some other options software, though, that leverages on TOS data. Avoid those because their IV modeling will be the same as TOS.
The only delta that is really correct is a crystal ball. Everything else is a guess especially when you are talking about otm options where vol resets can affect your delta as much as gamma.
I didn't read this entire thread but here are common reasons why you see different greeks reported from different vendors. The first thing is what are they using for their discount factor (i.e. risk-free rate)? Typically the way I go about doing this construct a yield a curve, then use appropriate values for given tenors. Now let's assume you're trying to price European options first. What inputs are they using? Typically you'll want to extract the implied forward from the options chain. Since you're doing that all of your carry, div, etc.. are implied within the forward, it's a risk-neutral input by definition, and there's no possibility of arbing the smile either. Oftentimes, vendors will simply use spot here which is incorrect. American gets a little more hairy in this regard, you can look at two different parts of the smile here. I'll leave that one up to you guys to figure out. There's also the issue of whether or not they're using a discrete dividend model or a continuous one. All of what I've described thus far is only in regards to how to estimate vol. Now to the actual computation of the greeks. Using the closed form BSM formulas is pretty straight forward. However if they're using finite differences then what's their step size?