I plan to begin my installation in the middle of my largest room and will use wood splines to reverse direction. I received my spline material yesterday and it looks very well made. It is 1/4” x 1/2” x 36”.

My question is about figment in the groove.

How tight or loose should the spline fit into the groove? Mine seems to have a little bit of play. I will use wood glue to keep the spline in place, and nail through the new tongue.

Any suggestions?

Finger tight. If you put the spline in and turn the board so that the spline is facing down and stays in place then you are good. Thin coat of glue so it doesn't ooze out and get all over the new flooring. If you glue and immediately nail put a a couple of pieces of flooring, with their grooves capturing the new spline, on each side of the nailer. These pieces will keep the spline in place and parallel to the subfloor and ensure that the floor board(s) that will be installed over the new spline will fit properly.

I was testing out the spline with a sample of the wood I will be getting. The spline is 1/4” x 1/2”. It fits but it seems sloppy. It will fall out if I turn the board over. Any suggestions?

Maybe I need to try a different brand of wood spline.

You could shim it to tighten the fit. But the concern would be the fit on the next row that fits over the spine....depending on the spline's location the boards may be able to move up/down. You could also cut your own splines.
Are the splines you bought a full 1/4" thick? How thick are the tongues on the flooring you are installing?
If you want to try and work with the splines you have I would locate them against the bottom of the groove. That way the next boards would be held flat to the subfloor. Try putting painter's tape on the top of the spline to see if this gives you a good fit. Don't be overly concerned about the amount of glue.....the flooring nails/staples provide the hold.

I think that the spline you have will work fine for keeping the flooring in line when you reverse direction, even though it will fall out because it is loose before nailing. Like Jimmie M has stated, when you are installing the spline, use a loose board next to where you are going to fasten the spline in place which will keep the spline from falling out as you nail or staple it into the groove of the board where you are fastening before you drive the fastener into place. The fastener will pull the slip spline down into position so it will hold the next board and the taper on both edges of the spline, which fits loosely into the groove of both boards will allow the next board to slide into position without any trouble.
The spline will be in the correct position to accept the next row of boards if it is fastened into place carefully.
After you fasten the first 3 foot length of spline into position you can be assured that the next row of boards are going to be held down with the spline even though it seems too loose before it is fastened.

Thanks for the advice. I tested with one piece of tape and it was snug. So perhaps that’s a good idea. I’m probably overthinking the fitment too much.

The flooring that will be installed is solid cherry 3-6” random width. I’ll be using 2” cleats and Wakol MS-262 glue to assist. Hopefully the flooring will be milled and shipped out within the next few weeks. I know acclimation is key.

You're not overthinking this too much. Joints that don't fit well are potential squeak spots. Unfortunately when you encounter these the flooring is already installed. Better to get it right the first time.