Presto lag dates, group/partitioned by id

Question

Say that I want to find every time that a client updated their budget.

Here's what my data looks like

datetime, client_id, new_budget
__________,__________,___________
2022-01-01,1,        ,100
2022-01-01,2,        ,300
2022-01-02,1,        ,80
2022-01-02,2,        ,80

And the code I've run.

SELECT datetime AS dt_1,
       LAG(datetime) OVER (ORDER BY client_id, datetime) AS dt_2,
       client_id,
       new_budget
FROM budget_table

What I'm expecting returned will be

dt_1,      dt_2,      client_id, new_budget
__________,__________,__________,___________
2022-01-01,NULL,      1         , 100
2022-01-02,2022-01-01,1         , 80
2022-01-01,NULL,      2         , 300
2022-01-02,2022-01-01,2         , 80 

Hence there are NULL values for dt_2 in the first entry of each client_id. I'm not sure what code will accomplish this effect; is a GROUP BY clause will be necessary (or a partition over clause.)

But here's the output to the SQL that I ran

dt_1,      dt_2,      client_id, new_budget
__________,__________,__________,___________
2022-01-01,NULL,      1         , 100
2022-01-02,2022-01-01,1         , 80
2022-01-01,2022-01-02,2         , 300
2022-01-02,2022-01-01,2         , 80 

So the huge issue here is that it's not recognizing that the dt_2 should be NULL if the previous row is from a different client_id.

Which syntax is recommended to accomplish this effect?