In my experience, for whatever that's worth - lol, I've almost always had trouble with leggy seedlings doing well afterwards. I've actually started new seeds after accidentally allowing seedlings to get too long and leggy and planted them all. The newer seedlings, even thought they were younger, quickly outgrew the older, yet weaker, leggy seedlings and produced more fruit. I'm not sure why it stunted them to get leggy but it did. It is important to keep the light right on top of the seedlings as they are growing because the strength of usable light disperses very quickly as you get further from the light source.
I've been using T5 fluorescent bulbs over the years and have been very happy with them. And now, last year and this year I added some LED shop lights to my seedling setup and I LOVE EM! Because they run so cool, I can drop the light right down on top of the seedlings and not worry about burning the plants. Not to mention they cost a fraction of the electricity to run. The initial cost of the LED shop light is quickly regained through energy savings and no more bulb replacements.
As for your reptile lights, they may run too hot. I wouldn't place my seedlings on a "precarious" windowsill, especially if cats are in the house but that's just me.
If you can keep the LED or fluorescent lights right down on top of your seedlings as soon as they sprout, they should not get leggy in the first place. On a side not, if any of your leggy plants are tomato plants by chance, then not to worry. You can always repot them in a taller pot or planter and bury the long leggy stem right up to the bottom set of leaves as the tiny hairs on the stem will then take root and thereby increase your root base. Only try this with tomatoes though as most other plants will not tolerate this.
Good luck with your seedlings and let us know how you make out.
*Haha - yeah what Chip said. It took me so long to type that out I didn't see that you already had the same answer - lol.