Rework query to include the Schema in the name - tsql

Question

I am leeching off this post: Query to list number of records in each table in a database
With this procedure:
CREATE PROCEDURE ListTableRowCounts
AS
BEGIN
SET NOCOUNT ON
CREATE TABLE #TableCounts
(
TableName VARCHAR(500),
CountOf INT
)
INSERT #TableCounts
EXEC sp_msForEachTable
'SELECT PARSENAME(''?'', 1),
COUNT(*) FROM ? WITH (NOLOCK)'
SELECT TableName , CountOf
FROM #TableCounts
ORDER BY TableName
DROP TABLE #TableCounts
END
GO
The procedure works well enough but I need it to output the name as Schema.Name and sort by that.
Is that possible? I'm not sure how to change this but you can see what it is doing below:
I have several instances were the table names are the same from different schemas.

Answer 1

CREATE PROCEDURE ListTableRowCounts
AS
BEGIN
SET NOCOUNT ON
CREATE TABLE #TableCounts
( SchemaName VARCHAR(500),
TableName VARCHAR(500),
CountOf INT
)
INSERT #TableCounts
EXEC sp_msForEachTable
'SELECT PARSENAME(''?'', 2), PARSENAME(''?'', 1),
COUNT(*) FROM ? WITH (NOLOCK)'
SELECT SchemaName, TableName , CountOf
FROM #TableCounts
ORDER BY TableName, SchemaName
DROP TABLE #TableCounts
END
GO

Answer 2

Taking some code from: https://stackoverflow.com/a/1443723/4584335
and from: How do I list all tables in all databases in SQL Server in a single result set?
Could I suggest this one (just in case "sp_msForEachTable" doesn't exist anymore):
declare #sql nvarchar(max);
select #sql = isnull(#sql + N'union all ', '')
+ N'
select b.name as "DB"
,a.name collate Latin1_General_CI_AI
,a.object_id
,a.schema_id
,' + cast(database_id as nvarchar(10)) + N'
,p.[Rows]
from ' + quotename(name) + N'.sys.tables a
join
' + quotename(name) + N'.sys.indexes i
on a.OBJECT_ID = i.object_id
and i.index_id <= 1
join
' + quotename(name) + N'.sys.partitions p
on i.object_id = p.OBJECT_ID
and i.index_id = p.index_id
join sys.databases b
on database_id=' + cast(database_id as nvarchar(10)) + ' '
from sys.databases
where state = 0
and user_access = 0;
exec sp_executesql #sql;