Help with window function

I am trying to count the number of distinct patients so far for any given day.

Here is a simplified example of what I am trying to achieve:

create table visit(datestring,timestring,patient);

insert into visit(datestring,timestring,patient) values ('2021-09-01','09:00','A');

insert into visit(datestring,timestring,patient) values ('2021-09-01','12:00','A');

insert into visit(datestring,timestring,patient) values ('2021-09-01','10:00','B');

insert into visit(datestring,timestring,patient) values ('2021-09-01','11:00','B');

insert into visit(datestring,timestring,patient) values ('2021-09-02','09:00','D');

insert into visit(datestring,timestring,patient) values ('2021-09-02','12:00','C');

insert into visit(datestring,timestring,patient) values ('2021-09-02','10:00','B');

insert into visit(datestring,timestring,patient) values ('2021-09-02','13:00','B');

which gives:

select * from visit order by datestring,timestring;

datestring	timestring	patient
2021-09-01	09:00	A
2021-09-01	10:00	B
2021-09-01	11:00	B
2021-09-01	12:00	A
2021-09-02	09:00	D
2021-09-02	10:00	B
2021-09-02	12:00	C
2021-09-02	13:00	B

The closest I could go to the expected result is here:

select datestring,timestring,patient,(case when count() over (partition by datestring,patient order by timestring rows between unbounded preceding and current row) > 1 then 0 else 1 end) as distinct_count from visit order by datestring,timestring;

which gives:

datestring	timestring	patient	distinct_count
2021-09-01	09:00	A	1
2021-09-01	10:00	B	1
2021-09-01	11:00	B	0
2021-09-01	12:00	A	0
2021-09-02	09:00	D	1
2021-09-02	10:00	B	1
2021-09-02	12:00	C	1
2021-09-02	13:00	B	0

How do I print the sum of the distinct_count over all the preceding rows which have the same datestring (including the current row)? It would this:

datestring	timestring	patient	distinct_count	what_I_want
2021-09-01	09:00	A	1	1
2021-09-01	10:00	B	1	2
2021-09-01	11:00	B	0	2
2021-09-01	12:00	A	0	2
2021-09-02	09:00	D	1	1
2021-09-02	10:00	B	1	2
2021-09-02	12:00	C	1	3
2021-09-02	13:00	B	0	3

I have spent a lot of time trying to figure this out. Any help or pointer would be greatly appreciated. Thank you.