First is a holly, second does not look like a spindle to me. But I can't say what it is instead.

#1 Yaupon holly

#2 perhaps variegated Ligustrum sinense?

Yes, to Dave's identifications. The berries on the Yaupon will soon turn a lustrous red, very ornamental.

I'm not totally convinced by the Ligustrum id. The leaves appear to be rather soft and undulating and the new stems appear to be a pale, soft green unlike any privet I know of.

1: Yaupon Holly

2: Euonymus Silver Queen

I would like to see some indication of the size of the leaves. Just looking at the photo they appear to be too small for Euonymus fortunei. Unless the colour of the picture is way off they also look far too yellow to be Silver Queen, which anyway is a climber and wouldn't be that tall unsupported. More photos would be very helpful.

I doubt #2 is any kind of euonymus. The texture is too fine for either Silver Queen or Silver King (both selections of E. japonicus, which is an upright growing shrub) and while it could be texturally compatible with a variegated form of fortunei (like Emerald Gaiety), that is a much more lax shrub or even with the climbing tendencies Floral referred to. If without a support on which to climb, it tend to form a large rounded but not upright shrub.

Like embothrium, it just doesn't seem very spindle like to me but I am not sure as to what else specifically. Have we completely ruled out variegated Chinese privet? Looks more like that than anything else I can think of and it's a very popular (if invasive) landscape plant in the Southeastern US.

If not sinense, it could be a Ligustrum hybrid, I suppose. Like L. x ibolium 'Variegatum'.

According to the RHS, and all the other sources I've quickly checked on, Euonymus Silver Queen is a fortunei. Euonymus SIlver Queen

I'd still like more pictures. But I am leaning towards Ligustrum sinense 'Variegatum' after all.

Although I am not sure why it would be different across the pond, 'Silver Queen' in this country is most typically a cultivar of E. japonicus. I do see where several sources include a fortunei cultivar with the same name, however 9 times out of 10, asking for or referring to that cultivar in the US will get you a Euonymus japonicus shrub.

I have never seen a fortunei cultivar sold under that name in this country.

Yes, I am pretty sure it is euonymus japonicus.

'Silver Queen' and 'Silver King' are similar and are japonicus cultivars in this country.

However, the OP's #2 plant is not Euonymus.

Euonymus has glossy leaves with some substance, and I know it's hard to tell, but these leaves don't look that way to me. And Euonymus leaves have a crenate leaf margin. Again, just don't see it.

And they are larger than what I see in the pic, although it is a bit hard to tell leaf size in the photo.

I would also be interested in knowing whether #2 plant is semi-deciduous or completely evergreen, although the further south one lives, the more L. sinense tends to be semi- to ever-green. Judging by the Indian Hawthorn in the photo, the OP is probably in zone 7b/8 or warmer.

It IS ABSOLUTELY variegated Ligustrum sinense. You wouldn't be confused by it if you could touch it.

I have seen L. sinense 'variegatum' in southern landscapes for years and years, much overused as a landscape foundation, hedge, screen, etc. It is often sheared, but has a natural arching growth habit.

Whitefly, by the clouds, is often a pest.

The OP is in Texas, according to the first posting. This ligustrum would typically be evergreen in much of Texas.

Ligustrum Sinese is evergreen here as well. If that where mine, I would rid myself of it before it blooms. Every time I start seeing privet blooming on roadsides, my horrible late spring allergies start. It's little baby abominations will pop up everywhere causing you more of a problem than it's worth.

It is in the top 6 of my do not plant list. I am going to make a more extensive one in the future.

==>>>https://caldwellhg.blogspot.com/2016/06/about-common-plants-of-south-you.html

Weird - Silver Queen - Bean - Silver Queen 1. - SQ2,

The chronically unstable 'Silver Queen' often seen here is an E. fortunei variant. Houtman, Variegated Trees & Shrubs - The Illustrated Encyclopedia (2004, Timber Press/The Royal Boskoop Horticultural Society) lists and describes both an E. fortunei 'Silver Queen' and an E. japonicus 'Silver Queen'.

Just reminding people that the informative discussion about the Euonymus nomenclature has nothing to do with the Ligustrum sinense in question :-)

I'm not convinced it's a ligustrum either.

Well, the OP has disappeared so maybe we'll never have any more information to work on.

It could be either, whether euonymus or ligustrum both can be problem plants. The size of this shrub is more ligustrum-like. I have never seen a euonymus japonicus that size. Either way, a chainsaw will be your best friend in this situation. Don't forget to but some strong herbicide, won't take ligustrum too long to regenerate.

Definitely not euonymus.