Rather than trying to define each Grouping as a new temp table I'd be tempted to do something more like for each account flagging if they made purchase in each year. Then using these flags to identify which group they should be in.

In this code I've left the individual flags as well as the CustomerGroup. I would use the flags to sanity check the logic on the grouping CASE.

SELECT DISTINCT txn.account_no,
                CASE WHEN txn1.account_no IS NOT NULL THEN 'Y' ELSE 'N' END AS Purchase1Year,
                CASE WHEN txn2.account_no IS NOT NULL THEN 'Y' ELSE 'N' END AS Purchase2Year,
                CASE WHEN txn3.account_no IS NOT NULL THEN 'Y' ELSE 'N' END AS Purchase3Year,
                CASE WHEN txn4.account_no IS NOT NULL THEN 'Y' ELSE 'N' END AS PurchaseOlder,
                CASE
                    WHEN txn1.account_no IS NULL AND txn2.account_no IS NOT NULL THEN 'Lost'
                    WHEN txn1.account_no IS NOT NULL AND txn2.account_no IS NULL AND txn3.account_no IS NULL AND txn4.account_no IS NULL THEN 'New'
                    WHEN txn1.account_no IS NOT NULL AND txn2.account_no IS NULL AND txn3.account_no IS NOT NULL THEN 'Restarted (1 year)'
                    WHEN txn1.account_no IS NOT NULL AND txn2.account_no IS NULL AND (txn3.account_no IS NOT NULL OR txn4.account_no IS NOT NULL) THEN 'Restarted (other)'
                    WHEN txn1.account_no IS NOT NULL AND txn2.account_no IS NOT NULL THEN 'Continuing'
                    ELSE 'other not captured'
                END AS CustomerGroup
FROM transactions AS txn
     LEFT JOIN transactions AS txn1 ON txn.account_no = txn.account_no AND txn1.tran_date BETWEEN DATEADD(DAY, -365, GETDATE()) AND GETDATE()
     LEFT JOIN transactions AS txn2 ON txn.account_no = txn.account_no AND txn2.tran_date BETWEEN DATEADD(DAY, -730, GETDATE()) AND DATEADD(DAY, -366, GETDATE())
     LEFT JOIN transactions AS txn3 ON txn.account_no = txn.account_no AND txn3.tran_date BETWEEN DATEADD(DAY, -1095, GETDATE()) AND DATEADD(DAY, -731, GETDATE())
     LEFT JOIN transactions AS txn4 ON txn.account_no = txn.account_no AND txn4.tran_date < DATEADD(DAY, -1095, GETDATE());

There is a customer group for other that under the current definition will include anyone who has not purchased in year 1 nor 2, but has made an older purchase. You may want to include this filter

WHERE txn1.account_no IS NOT NULL OR txn2.account_no IS NOT NULL

It's really helpful for questions like this to include sample DDL and DML. Based on your description I put together this as example data. I've used table variables because there's less to clean up, but that's just a personal preference.

DECLARE @Orders TABLE (OrderID BIGINT PRIMARY KEY IDENTITY, CustomerID BIGINT, OrderDate DATETIME);
DECLARE @Customers TABLE (CustomerID BIGINT PRIMARY KEY IDENTITY, FirstName NVARCHAR(20), LastName NVARCHAR(20));

INSERT INTO @Orders (CustomerID, OrderDate) VALUES
(1, '2023-02-01'),(1, '2022-02-01'),(1, '2021-02-01'),
(2, '2023-02-01'),
(3, '2023-02-01'),(3, '2021-06-01'),
(4, '2023-02-01'),(4, '2020-06-01'),
(5, '2015-06-01'),(5, '2023-02-01'),
(6, '2020-06-01');

INSERT INTO @Customers (FirstName, LastName) VALUES
('Pike','Trickfoot'),('Percival','de Rolo'),('Vax','Vessar'),('Vex','Vessar'),('Grog','Strongjaw'),('Scanlan','Shorthalt')

Using that, I then crafted a case expression to look over several LEFT OUTER joins to the orders which capture the specifics you talked about:

SELECT c.CustomerID, C.FirstName, C.LastName, COUNT(o36.OrderID) AS o36, COUNT(o24.OrderID) AS o24, COUNT(o1223.OrderID) AS o1223, COUNT(o012.OrderID) AS o012,
CASE WHEN COUNT(o36.OrderID) = 0 AND COUNT(o24.OrderID) > 0 AND COUNT(o1223.OrderID) = 0 AND COUNT(o012.OrderID) = 0 THEN 'Lost'
     WHEN COUNT(o36.OrderID) = 0 AND COUNT(o24.OrderID) = 0 AND COUNT(o1223.OrderID) = 0 AND COUNT(o012.OrderID) > 0 THEN 'New'
     WHEN COUNT(o36.OrderID) > 0 AND COUNT(o24.OrderID) = 0 AND COUNT(o1223.OrderID) = 0 AND COUNT(o012.OrderID) > 0 THEN 'Restarted (Other)'
     WHEN COUNT(o36.OrderID) = 0 AND COUNT(o24.OrderID) > 0 AND COUNT(o1223.OrderID) = 0 AND COUNT(o012.OrderID) > 0 THEN 'Restarted (1 Year)'
     WHEN COUNT(o1223.OrderID) > 0 AND COUNT(o012.OrderID) > 0 THEN 'Continuing'
END AS CustomerStatus
  FROM @Customers c
    LEFT OUTER JOIN @Orders o36
      ON c.CustomerID = o36.CustomerID
      AND o36.OrderDate < DATEADD(MONTH,-36,CURRENT_TIMESTAMP)
    LEFT OUTER JOIN @Orders o24
      ON c.CustomerID = o24.CustomerID
      AND o24.OrderDate BETWEEN DATEADD(MONTH,-36,CURRENT_TIMESTAMP) AND DATEADD(MONTH,-24,CURRENT_TIMESTAMP)
    LEFT OUTER JOIN @Orders o1223
      ON c.CustomerID = o1223.CustomerID
      AND o1223.OrderDate BETWEEN DATEADD(MONTH,-24,CURRENT_TIMESTAMP) AND DATEADD(MONTH,-12,CURRENT_TIMESTAMP)
    LEFT OUTER JOIN @Orders o012
      ON c.CustomerID = o012.CustomerID
      AND o012.OrderDate > DATEADD(MONTH,-12,CURRENT_TIMESTAMP) 
 GROUP BY c.CustomerID, C.FirstName, C.LastName

We're joining to the orders table several times for the periods you mention, and then just doing a simple count for each of the periods. The case expression then returns the status dependent on what's found.

You may want to tweak exactly how the joins work to tweak it to better fit your needs.