1) Saul died.
2) Saul’s three sons died.
3) Saul’s house died.

Samuel says:

1) Saul died.
2) Saul’s three sons died.
3) Saul’s men died.

“Saul’s house” refers to the same group that “Saul’s men” refers to: the group of Saul’s hirelings and retainers. In this context, I think it only refers to those of Saul’s hirelings that were present at the battle. Presumably, the passage did not intend to say that the cook in the palace died in this battle as well. The NASB captures this by translating the passage “all those of his house” where the NRSV says “all his house”.

Esh-Baal is not intended, by either Samuel or the Chronicler, to be among that group. Esh-Baal was a son, not a mere household man. Had the writer of either work intended to include Esh-Baal among the dead, he would have described his death as he did the other three sons, and said that Saul’s four sons had died.