I have to disagree with this whole "train the tenant" approach. Granted, boundaries are important, but let's remember that tenants are people.
This is your first-test run of this lease clause, and as it turns out, it's probably not a great lease clause, and you're finding that out.
Now, in your life, do you mow your lawn every Sunday, no matter what? Do you never get busy? Does it never rain?
I think the problem is this lease clause. I don't think it's realistic. And if your tenant is set up to fail, then he will. If you get all militant on him, he'll be hostile and you can bet he'll leave at the end of the lease.
You are immensely blessed to have a tenant, especially in your first rental, who pays on time and does not have any other issues. And, he's obviously a hard worker.
So, the only issue you have here, is a great tenant who is working too many hours to mow the lawn.
I always advise landlords to include the cost of yard maintenance in their rents. Tenants will never keep the yard up the way you would like them to. It's an important way to protect your investment. It also gives you a reason to be on the property once in a while - while you're there to meet the gardener. Unobtrusively, of course :-)
I also think leaving a lawn mower with a tenant is a bad idea, because it could get stolen or abused, and that will be an issue to deal with.
So, how much would the lawn service cost? If it's not horrendous, I'd be tempted to just take back the lawn mower and pay a service, and just eat the cost. Let the tenant know when they'll be showing up. That you value them as a tenant, and can see that the lawncare is going to be too much for them, so you're going to just take care of it.
Option B: Tell them you'll do the above if they sign another year lease. That you'll keep the rent the same, but you want a two year lease, and you will provide a gardener.
If you try to get them to pay for a gardener now, they'll balk most likely. They'll want to save that money, and will then try to get the lawn mowed. They'll be stressing over it, you won't be happy, and they'll be discussing how they need to rent a place that doesn't require them to do yard work.
So, if there's any way you can just provide the service, that's my recommendation.
Figure the cost of turning over that unit at the end of December, if they don't renew. Yep, middle of winter. Probably at least one month's lost rent even in the best weather, with painting and cleaning, etc., and lost rent between tenants. So, would the cost of the lawn service add up to a month's rent? At least?
Tenants can "feel" when a landlord is trying to "train" them. This isn't about training people. Tenants are not serfs. This is a business relationship. And yes, a contract is a contract. But sometimes a new contract is entered into with the best intentions of both parties, and it just isn't working. So, look for a solution, not a punishment, when you're dealing with an otherwise fantastic tenant. In my opinion.
And include lawn service in the rent in the next lease. :-)