Mower
For 13 acres there is only one type you should consider that is a rear mounted rotary on a small tractor
A well reconditioned MF 35 or 135 should cost £3000 or so and a good 2.4m rotary topper from £1500 up S/H or £3500 new if you keep them in good repair you will sell it for more when the time comes.
Don't worry about running over the grass, mow weekly and it will be a lovely finish, the beauty of a rotary is that you can mow wet grass, even in the rain if you need to, just keep the blades sharp.
Don't consider a cylinder type the only work on dry grass and a flail is slower and uses more power.
You might decide a more modern tractor was more desirable, it may not cost much more to buy - but, may cost a lot more to repair, a small JD is particularly complex and expensive.