Depends what you want.
They have admitted they acted wrongly and have apologised for it.
As I mentioned before, should you want to bring a case for unfair dismissal at a tribunal you would have to go through an internal grievance procedure first. Often when the person has left employment already, as you have, you would use the modified procedure as they have described. Now that you've raised a grievance they are giving you the option to do this or to go through the full procedure.
It's completely up to you. If you wanted to take it further (bearing in mind all we've discussed above) you will need to decide which procedure to use, modified one easiest, and let them know.
I think all you probably want is to get a decent reference agreed and to ensure that when supplying information to potential future employers, they state that you resigned not that you were dismissed.
If I were you I'd probably write to them (by email for speed but consider a letter as well) thanking them for their quick response and for the apology for their conduct.
State that you would like written assurance that information provided to future employers in the form of a reference will refer to your employment ending by resignation not dismissal. Following your dismissal you are also naturally concerned about what comments would be put on a future reference in terms of your work capabilities as well. You would also like to agree with them appropriate wording for a more detailed reference with regard to your work to be supplied to future employers should that be required.
if they indicate that they are happy to provide both the written assurance and to agree appropriate wording for a more detailed work reference, you will be happy to confirm that you will drop the grievance.
Should they not be happy with these requests, you will proceed with the grievance and would [be happy to use the modified procedure][like a grievance meeting in line with the standard grievance procedure] (delete as appropriate).
Advantages of requesting the full procedure including a meeting would be that you would be able to discuss your concerns about references in person and might get the reassurances/agreement you want quite easily.
What you could do is write along the lines I've suggested, giving the standard grievance procedure as an alternative if they do not agree straight-off to the things you've requested, then you will get an opportunity to discuss your concerns.
I think by giving a written apology straight away they are probably hoping you'll drop it now as by bringing a case you are not really going to get anything more. But you need to make it clear that there are a couple of things that need ironing out before that.